* Liu et al. (2019) Jiaying Liu, Xiangjie Kong, Xinyu Zhou, Lei Wang, Da Zhang, Ivan Lee, Bo Xu, and Feng Xia. 2019.  Data Mining and Information Retrieval in the 21st Century: A Bibliographic Review.  _Computer science review_ 34 (2019), 100193.
  * Ma et al. (2020) Pingchuan Ma, Shuai Wang, and Jin Liu. 2020.  Metamorphic Testing and Certified Mitigation of Fairness Violations in NLP Models. In _Proceedings of the 2020 International Joint Conference on Artificial Intelligence_. Morgan Kaufmann, 458–465.
  * Madhusoodanan (2020) Jyoti Madhusoodanan. 2020\.  Is a Racially-biased Algorithm Delaying Health Care for One Million Black People?  _Nature_ (2020), 546–547.
  * Maheshwari and Perrot (2023) Gaurav Maheshwari and Michaël Perrot. 2023.  FairGrad: Fairness Aware Gradient Descent.  _Transactions on Machine Learning Research_ (2023).
  * Mallari et al. (2020) Keri Mallari, Kori Inkpen, Paul Johns, Sarah Tan, Divya Ramesh, and Ece Kamar. 2020\.  Do I Look Like a Criminal? Examining how Race Presentation Impacts Human Judgement of Recidivism. In _Proceedings of the ACM Conference on Human Factors in Computing Systems_. ACM, 1–13.
  * Mancini et al. (2022) Anna Mancini, Antonio Desiderio, Riccardo Di Clemente, and Giulio Cimini. 2022.  Self-induced Consensus of Reddit Users to Characterise the GameStop Short Squeeze.  _Scientific Reports_ 12, 1 (2022), 13780.
  * McInerney et al. (2020) James McInerney, Brian Brost, Praveen Chandar, Rishabh Mehrotra, and Benjamin Carterette. 2020\.  Counterfactual Evaluation of Slate Recommendations with Sequential Reward Interactions. In _Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_. ACM, 1779–1788.
  * Mehrotra and Vishnoi (2022) Anay Mehrotra and Nisheeth Vishnoi. 2022.  Fair Ranking with Noisy Protected Attributes.  _Proceedings of the 36th Conference on Neural Information Processing Systems_ , 31711–31725.
  * Metevier et al. (2019) Blossom Metevier, Stephen Giguere, Sarah Brockman, Ari Kobren, Yuriy Brun, Emma Brunskill, and Philip Thomas. 2019.  Offline Contextual Bandits with High Probability Fairness Guarantees. In _Proceedings of the 33th Conference on Neural Information Processing Systems_. MIT Press.
  * Michela et al. (2016) Del Vicario Michela, Vivaldo Gianna, Bessi Alessandro, Zollo Fabiana, Scala Antonio, Caldarelli Guido, and Quattrociocchi Walter. 2016.  Echo Chambers: Emotional Contagion and Group Polarization on Facebook.  _Scientific Reports_ (2016), 37825\.
  * Miklós-Thal and Tucker (2019) Jeanine Miklós-Thal and Catherine Tucker. 2019.  Collusion by algorithm: does better demand prediction facilitate coordination between sellers?  _Management Science_ 65, 4 (2019), 1552–1561.
  * Mitchell et al. (2020) Shira Mitchell, Eric Potash, Solon Barocas, Alexander D’Amour, and Kristian Lum. 2020\.  Algorithmic Fairness: Choices, Assumptions, and Definitions.  _Annual Review of Statistics and Its Application_ 8 (2020), 141–163.
  * Mladenov et al. (2020) Martin Mladenov, Elliot Creager, Omer Ben-Porat, Kevin Swersky, Richard Zemel, and Craig Boutilier. 2020\.  Optimizing Long-term Social Welfare in Recommender Systems: A Constrained Matching Approach. In _Proceedings of the 2020 International Conference on Machine Learning_. ACM, 6987–6998.
  * Munn (2020) Luke Munn. 2020\.  Angry by Design: Toxic Communication and Technical Architectures.  _Humanities and Social Sciences Communications_ 7, 1 (2020), 1–11.
  * Nabi and Shpitser (2018) Razieh Nabi and Ilya Shpitser. 2018.  Fair Inference on Outcomes. In _Proceedings of the 2018 AAAI Conference on Artificial Intelligence_ , Vol. 32. AAAI.
  * Niu et al. (2020) Xichuan Niu, Bofang Li, Chenliang Li, Rong Xiao, Haochuan Sun, Hongbo Deng, and Zhenzhong Chen. 2020.  A Dual Heterogeneous Graph Attention Network to Improve Long-tail Performance for Shop Search in E-commerce. In _Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_. ACM, 3405–3415.
  * Ntoutsi et al. (2020) Eirini Ntoutsi, Pavlos Faflios, Ujwal Gadiraju, Vasileios Iosifidis, Wolfgang Nejdl, Maria-Esther Vidal, Salvatore Ruggieri, Franco Turini, Symeon Papadopoulos, Emmanouil Krasanakis, Ioannis Kompatsiaris, Katharina Kinder-Kurlanda, Claudia Wagner, Fariba Karimi, Miriam Fernandez, Harith Alani, Bettina Berendt, Tina Kruegel, Christian Heinze, Klaus Broelemann, Gjergji Kasneci, Thanassis Tiropanis, and Steffen Staab. 2020.  Bias in Data-driven Artificial Intelligence Systems \- An Introducutory Survey.  _Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery_ 10, 1 (2020), 115:1–115:35.
  * Obermeyer et al. (2019) Ziad Obermeyer, Brian Powers, Christine Vogeli, and Sendhil Mullainathan. 2019.  Dissecting Racial Bias in an Algorithm Used to Manage the Health of Populations.  _Science_ 366, 6464 (2019), 447–453.
  * Oneto et al. (2019) Luca Oneto, Michele Doninini, Amon Elders, and Massimiliano Pontil. 2019.  Taking Advantage of Multitask Learning for Fair Classification. In _Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society_. ACM, 227–237.
  * Otterbacher et al. (2018) Jahna Otterbacher, Alessandro Checco, Gianluca Demartini, and Paul Clough. 2018.  Investigating User Perception of Gender Bias in Image Search: The Role of Sexism. In _Proceedings of the 2018 International ACM SIGIR Conference on Research & Development in Information Retrieval_. ACM, 933–936.
  * Oyku and Shen (2022) Deniz Kose Oyku and Yanning Shen. 2022.  Fair Contrastive Learning on Graphs.  _IEEE Transactions on Signal and Information Processing over Networks_ 8 (2022), 475–488.
  * Pariser (2011) Eli Pariser. 2011\.  _The Filter Bubble: What the Internet is Hiding from You_.  Penguin UK.
  * Peng et al. (2019) Andi Peng, Besmira Nushi, Emre Kıcıman, Kori Inkpen, Siddharth Suri, and Ece Kamar. 2019\.  What You See is What You Get? The Impact of rRepresentation Criteria on Human Bias in Hiring. In _Proceedings of the 2019 AAAI Conference on Human Computation and Crowdsourcing_ , Vol. 7. AAAI, 125–134.
  * Pennington et al. (2014) Jeffrey Pennington, Richard Socher, and Christopher D Manning. 2014.  Glove: Global vectors for word representation. In _Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing_. ACL, 1532–1543.
  * Pessach and Shmueli (2022) Dana Pessach and Erez Shmueli. 2022.  A Review on Fairness in Machine Learning.  _Comput. Surveys_ 55, 3 (2022), 51:1–51:44.
  * Piano (2020) Samuele Lo Piano. 2020\.  Ethical Principles in Machine Learning and Artificial Intelligence: Cases from the Field and Possible Ways Forward.  _Humanities and Social Sciences Communications_ 7, 1 (2020), 1–7.
  * Pierri et al. (2020) Francesco Pierri, Carlo Piccardi, and Stefano Ceri. 2020\.  Topology Comparison of Twitter Diffusion Networks Effectively Reveals Misleading Information.  _Scientific Reports_ 10, 1 (2020), 1–9.
  * Poria et al. (2020) Soujanya Poria, Devamanyu Hazarika, Navonil Majumder, and Rada Mihalcea. 2020.  Beneath the Tip of the Iceberg: Current Challenges and New Directions in Sentiment Analysis Research.  _IEEE Transactions on Affective Computing_ (2020), 1–1.
  * Preoţiuc-Pietro et al. (2017) Daniel Preoţiuc-Pietro, Ye Liu, Daniel Hopkins, and Lyle Ungar. 2017.  Beyond Binary Labels: Political Ideology Prediction of Twitter Users. In _Proceedings of the 2017 Annual Meeting of the Association for Computational Linguistics_. ACL.
  * Qin et al. (2020) Zhen Qin, Suming J Chen, Donald Metzler, Yongwoo Noh, Jingzheng Qin, and Xuanhui Wang. 2020\.  Attribute-based Propensity for Unbiased Learning in Recommender Systems: Algorithm and Case Studies. In _Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_. ACM, 2359–2367.
  * Rai (2019) Tage S. Rai. 2019\.  Racial Bias in Health Algorithms.  _Science_ 366 (2019), 440–441.
  * Raj and Ekstrand (2022) Amifa Raj and Michael D Ekstrand. 2022.  Measuring Fairness in Ranked Results: An Analytical and Empirical Comparison. In _Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval_. ACM, 726–736.
  * Rhee et al. (2022) Wondo Rhee, Sung Min Cho, and Bongwon Suh. 2022.  Countering Popularity Bias by Regularizing Score Differences. In _Proceedings of the 16th ACM Conference on Recommender Systems_. ACM, 145–155.
  * Roberts et al. (2021) Michael Roberts, Derek Driggs, Matthew Thorpe, Julian Gilbey, Michael Yeung, Stephan Ursprung, Angelica I Aviles-Rivero, Christian Etmann, Cathal McCague, Lucian Beer, et al. 2021\.  Common Pitfalls and Recommendations for Using Machine Learning to Detect and Prognosticate for COVID-19 Using Chest Radiographs and CT Scans.  _Nature Machine Intelligence_ 3, 3 (2021), 199–217.
  * Robertson et al. (2018) Ronald E Robertson, David Lazer, and Christo Wilson. 2018\.  Auditing the Personalization and Composition of Politically-related Search Engine Results Pages. In _Proceedings of the ACM Web Conference_. ACM, 955–965.
  * Roselli et al. (2019) Drew Roselli, Jeanna Matthews, and Nisha Talagala. 2019\.  Managing Bias in AI. In _Proceedings of the ACM Web Conference_. ACM, 539–544.
  * Rueden et al. (2023) Laura Von Rueden, Sebastian Mayer, Katharina Beckh, Bogdan Georgiev, Sven Giesselbach, Raoul Heese, Birgit Kirsch, Julius Pfrommer, Annika Pick, Rajkumar Ramamurthy, et al. 2023\.  Informed Machine Learning–a Taxonomy and Survey of Integrating Prior Knowledge into Learning Systems.  _IEEE Transactions on Knowledge and Data Engineering_ 35, 1 (2023), 614–633.
  * Saeid et al. (2022) Tizpaz-Niari Saeid, Kumar Ashish, Tan Gang, and Trivedi Ashuotosh. 2022.  Fairness-aware Configuration of Machine Learning Libraries. In _Proceedings of the 44th International Conference on Software Engineering_. ACM/IEEE, 909–920.
  * Saito (2020) Yuta Saito. 2020\.  Unbiased Pairwise Learning from Biased Implicit Feedback. In _Proceedings of the 2020 International ACM SIGIR Conference on Theory of Information Retrieval_. ACM, 5–12.
  * Saito and Joachims (2022) Yuta Saito and Thorsten Joachims. 2022.  Fair Ranking as Fair Division: Impact-Based Individual Fairness in Ranking. In _Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining_. ACM, 1514–1524.
  * Salimi et al. (2020) Babak Salimi, Bill Howe, and Dan Suciu. 2020.  Database Repair Meets Algorithmic Fairness.  _ACM SIGMOD Record_ 49, 1 (2020), 34–41.
  * Sanford and Lorimer (2022) Mary Sanford and Jamie Lorimer. 2022.  Veganuary and the Vegan Sausage (T)rolls: Conflict and Commercial Engagement in Online Climate-diet Discourse.  _Humanities and Social Sciences Communications_ 9, 1 (2022), 1–13.
  * Sap et al. (2019) Maarten Sap, Dallas Card, Saadia Gabriel, Yejin Choi, and Noah A Smith. 2019.  The Risk of Racial Bias in Hate Speech Detection. In _Proceedings of the 2019 Annual Meeting of the Association for Computational Linguistics_. ACL, 1668–1678.
  * Saunders and Byrne (2020) Danielle Saunders and Bill Byrne. 2020.  Reducing Gender Bias in Neural Machine Translation as a Domain Adaptation Problem. In _Proceedings of the 2020 Annual Meeting of the Association for Computational Linguistics_. ACL.
  * Schor and Vallas (2021) Juliet B Schor and Steven P Vallas. 2021.  The Sharing Economy: Rhetoric and Reality.  _Annual Review of Sociology_ 47 (2021).
  * Sikder et al. (2020) Orowa Sikder, Robert E Smith, Pierpaolo Vivo, and Giacomo Livan. 2020.  A Minimalistic Model of Bias, Polarization and Misinformation in Social Networks.  _Scientific Reports_ 10, 1 (2020), 1–11.
  * Silva and Kenney (2019) Selena Silva and Martin Kenney. 2019.  Algorithms, Platforms, and Ethnic Bias.  _Commun. ACM_ 62, 11 (2019), 37–39.
  * Singh and Joachims (2019) Ashudeep Singh and Thorsten Joachims. 2019.  Policy Learning for Fairness in Ranking. In _Proceedings of the 33th Conference on Neural Information Processing Systems_. MIT Press, 5427–5437.
  * Smith et al. (2023) Jessie J Smith, Anas Buhayh, Anushka Kathait, Pradeep Ragothaman, Nicholas Mattei, Robin Burke, and Amy Voida. 2023.  The Many Faces of Fairness: Exploring the Institutional Logics of Multistakeholder Microlending Recommendation. In _Proceedings of the 2023 ACM Conference on Fairness, Accountability, and Transparency_. ACM, 1652–1663.
  * Stanovsky et al. (2019) Gabriel Stanovsky, Noah A Smith, and Luke Zettlemoyer. 2019\.  Evaluating Gender Bias in Machine Translation. In _Proceedings of the 2020 Annual Meeting of the Association for Computational Linguistics_ , Vol. 1. 1679–1684.
  * Stoica and Chaintreau (2019) Ana-Andreea Stoica and Augustin Chaintreau. 2019.  Hegemony in Social Media and the Effect of Recommendations. In _Proceedings of the ACM Web Conference_. ACM, 575–580.
  * Stoica et al. (2020) Ana-Andreea Stoica, Jessy Xinyi Han, and Augustin Chaintreau. 2020.  Seeding Network Influence in Biased Networks and the Benefits of Diversity. In _Proceedings of the ACM Web Conference_. ACM, 2089–2098.
  * Stoica et al. (2018) Ana-Andreea Stoica, Christopher Riederer, and Augustin Chaintreau. 2018.  Algorithmic Glass Ceiling in Social Networks: The Effects of Social Recommendations on Network Diversity. In _Proceedings of the ACM Web Conference_. ACM, 923–932.
  * Sun et al. (2019) Wenlong Sun, Sami Khenissi, Olfa Nasraoui, and Patrick Shafto. 2019.  Debiasing the Human-recommender System Feedback Loop in Collaborative Filtering. In _Proceedings of the ACM Web Conference_. ACM, 645–651.
  * Sweeney and Najafian (2019) Chris Sweeney and Maryam Najafian. 2019.  A Transparent Framework for Evaluating Unintended Demographic Bias in Word Embeddings. In _Proceedings of the 2019 Annual Meeting of the Association for Computational Linguistics_. ACL, 1662–1667.
  * Tan et al. (2020) Samson Tan, Shafiq Joty, Min-Yen Kan, and Richard Socher. 2020\.  It’s Morphin’Time! Combating Linguistic Discrimination with Inflectional Perturbations. In _Proceedings of the 2020 Annual Meeting of the Association for Computational Linguistics_. ACL, 2920–2935.
  * Tien et al. (2014) T.Nguyen Tien, Hui Pik-mai, Harper F.Maxwell, Terveen Loren, and A.konstan Joseph. 2014\.  Exploring the Filter Bubble: the Effect of Using Recommender Systems on Content Diversity. In _Proceedings of the ACM Web Conference_. ACM, 677–686.
  * Uddin et al. (2023) Md Taufeeq Uddin, Lijun Yin, and Shaun Canavan. 2023\.  Spatio-Temporal Graph Analytics on Secondary Affect Data for Improving Trustworthy Emotional AI.  _IEEE Transactions on Affective Computing_ (2023).
  * Vaidya et al. (2020) Ameya Vaidya, Feng Mai, and Yue Ning. 2020.  Empirical Analysis of Multi-task Learning for Reducing Identity Bias in Toxic Comment Detection. In _Proceedings of the 2020 AAAI Conference on Web and Social Media_ , Vol. 14. AAAI, 683–693.
  * Wan et al. (2019) Liangtian Wan, Yuyuan Yuan, Feng Xia, and Huan Liu. 2019\.  To Your Surprise: Identifying Serendipitous Collaborators.  _IEEE Transactions on Big Data_ (2019).
  * Wan et al. (2023) Mingyang Wan, Daochen Zha, Ninghao Liu, and Na Zou. 2023\.  In-processing Modeling Techniques for Machine Learning Fairness: A survey.  _ACM Transactions on Knowledge Discovery from Data_ 17, 3 (2023), 1–27.
  * Wang et al. (2019b) Jinzhong Wang, Xiangjie Kong, Feng Xia, and Lijun Sun. 2019b.  Urban Human Mobility: Data-driven Modeling and Prediction.  _ACM SIGKDD Explorations Newsletter_ 21, 1 (2019), 1–19.
  * Wang et al. (2022) Jun Wang, Benjamin Rubinstein, and Trevor Cohn. 2022\.  Measuring and Mitigating Name Biases in Neural Machine Translation. In _Proceedings of the 2022 Annual Meeting of the Association for Computational Linguistics_. ACL, 2576–2590.
  * Wang et al. (2019a) Mei Wang, Weihong Deng, Jiani Hu, Xunqiang Tao, and Yaohai Huang. 2019a.  Racial Faces in the Wild: Reducing Racial Bias by Information Maximization Adaptation Network. In _Proceedings of the 2019 International Conference on Computer Vision_. IEEE, 692–702.
  * Wang et al. (2021b) Nan Wang, Zhen Qin, Xuanhui Wang, and Hongning Wang. 2021b.  Non-Clicks Mean Irrelevant? Propensity Ratio Scoring As a Correction. In _Proceedings of the 2021 ACM International Conference on Web Search and Data Mining_. ACM, 481–489.
  * Wang et al. (2019c) Qinyong Wang, Hongzhi Yin, Hao Wang, Quoc Viet Hung Nguyen, Zi Huang, and Lizhen Cui. 2019c.  Enhancing Collaborative Filtering with Generative Augmentation. In _Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_. ACM, 548–556.
  * Wang et al. (2020a) Tianlu Wang, Xi Victoria Lin, Nazneen Fatema Rajani, Bryan McCann, Vicente Ordonez, and Caiming Xiong. 2020a.  Double-hard Debias: Tailoring Word Embeddings for Gender Bias Mitigation. In _Proceedings of the 2020 Annual Meeting of the Association for Computational Linguistics_. ACL.
  * Wang et al. (2021a) Wenjie Wang, Fuli Feng, Xiangnan He, Liqiang Nie, and Tat-Seng Chua. 2021a.  Denoising Implicit Feedback for Recommendation. In _Proceedings of the 2021 ACM International Conference on Web Search and Data Mining_. ACM, 373–381.
  * Wang et al. (2020b) Wei Wang, Tao Tang, Feng Xia, Zhiguo Gong, Zhikui Chen, and Huan Liu. 2020b.  Collaborative Filtering with Network Representation Learning for Citation Recommendation.  _IEEE Transactions on Big Data_ (2020).
  * Wang et al. (2021c) Xiaojie Wang, Rui Zhang, Yu Sun, and Jianzhong Qi. 2021c.  Combating Selection Biases in Recommender Systems with a Few Unbiased Ratings. In _Proceedings of the 2021 ACM International Conference on Web Search and Data Mining_. ACM, 427–435.
  * Wang et al. (2023) Yifan Wang, Weizhi Ma, Min Zhang, Yiqun Liu, and Shaoping Ma. 2023.  A Survey on the Fairness of Recommender Systems.  _ACM Transactions on Information Systems_ 41, 3 (2023), 1–43.
  * Webster et al. (2018) Kellie Webster, Marta Recasens, Vera Axelrod, and Jason Baldridge. 2018.  Mind the Gap: A Balanced Corpus of Gendered Ambiguous Pronouns.  _Transactions of the Association for Computational Linguistics_ 6 (2018), 605–617.
  * Wick et al. (2019) Michael Wick, Swetasudha Panda, and Jean-Baptiste Tristan. 2019\.  Unlocking Fairness: a Trade-off Revisited. In _Proceedings of the 33th Conference on Neural Information Processing Systems_. MIT Press, 8780–8789.
  * Wu et al. (2019) Qiong Wu, Yong Liu, Chunyan Miao, Binqiang Zhao, Yin Zhao, and Lu Guan. 2019\.  PD-GAN: Adversarial Learning for Personalized Diversity-promoting Recommendation.. In _Proceedings of the 2019 International Joint Conference on Artificial Intelligence_ , Vol. 19. Morgan Kaufmann, 3870–3876.
  * Wu et al. (2021) Xinwei Wu, Hechang Chen, Jiashu Zhao, Li He, Dawei Yin, and Yi Chang. 2021\.  Unbiased Learning to Rank in Feeds Recommendation. In _Proceedings of the 2021 ACM International Conference on Web Search and Data Mining_. ACM, 490–498.
  * Xia et al. (2022) Feng Xia, Teng Guo, Xiaomei Bai, Adrian Shatte, Zitao Liu, and Jiliang Tang. 2022\.  SUMMER: Bias-aware Prediction of Graduate Employment Based on Educational Big Data.  _ACM/IMS Transactions on Data Science_ 2, 4 (2022), 1–24.
  * Xia et al. (2021) Feng Xia, Ke Sun, Shuo Yu, Abdul Aziz, Liangtian Wan, Shirui Pan, and Huan Liu. 2021.  Graph Learning: A Survey.  _IEEE Transactions on Artificial Intelligence_ 2, 2 (2021), 109–127.
  * Xiao et al. (2020) Zhiping Xiao, Weiping Song, Haoyan Xu, Zhicheng Ren, and Yizhou Sun. 2020.  TIMME: Twitter Ideology-detection via Multi-task Multi-relational Embedding. In _Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_. ACM, 2258–2268.
  * Xu et al. (2022) Chengpei Xu, Wenjing Jia, Ruomei Wang, Xiangjian He, Baoquan Zhao, and Yuanfang Zhang. 2022\.  Semantic Navigation of PowerPoint-Based Lecture Video for AutoNote Generation.  _IEEE Transactions on Learning Technologies_ 16, 1 (2022), 1–17.
  * Xu et al. (2019) Chengpei Xu, Ruomei Wang, Shujin Lin, Xiaonan Luo, Baoquan Zhao, Lijie Shao, and Mengqiu Hu. 2019.  Lecture2note: Automatic generation of lecture notes from slide-based educational videos. In _2019 IEEE International Conference on Multimedia and Expo (ICME)_. IEEE, 898–903.
  * Xu et al. (2020) Tian Xu, Jennifer White, Sinan Kalkan, and Hatice Gunes. 2020\.  Investigating Bias and Fairness in Facial Expression Recognition. In _Proceedings of the 2020 European Conference on Computer Vision_. Springer, 506–523.
  * Yalcin and Bilge (2022) Emre Yalcin and Alper Bilge. 2022.  Evaluating Unfairness of Popularity Bias in Recommender Systems: A Comprehensive User-centric Analysis.  _Information Processing & Management_ 59, 6 (2022), 103100\.
  * Yang et al. (2023) Tao Yang, Zhichao Xu, Zhenduo Wang, Anh Tran, and Qingyao Ai. 2023.  Marginal-Certainty-aware Fair Ranking Algorithm. In _Proceedings of the 16th ACM International Conference on Web Search and Data Mining_. ACM, 24–32.
  * Ye and Skiena (2019) Junting Ye and Steven Skiena. 2019.  Mediarank: Computational Ranking of Online News Sources. In _Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_. ACM, 2469–2477.
  * Yin et al. (2020) Jianwen Yin, Chenghao Liu, Weiqing Wang, Jianling Sun, and Steven CH Hoi. 2020.  Learning Transferrable Parameters for Long-tailed Sequential User Behavior Modeling. In _Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_. ACM, 359–367.
  * Yu et al. (2021) Shuo Yu, Qing Qing, Chen Zhang, Ahsan Shehzad, Giles Oatley, and Feng Xia. 2021\.  Data-Driven Decision-Making in COVID-19 Response: A Survey.  _IEEE Transactions on Computational Social Systems_ 8, 4 (2021), 1016–1029.
  * Yu et al. (2022) Shuo Yu, Jing Ren, Shihao Li, Mehdi Naseriparsa, and Feng Xia. 2022.  Graph Learning for Fake Review Detection.  _Forntiers in Artificial Intelligence_ 5 (2022), 922589.
  * Yuan et al. (2020) Bowen Yuan, Yaxu Liu, Jui-Yang Hsia, Zhenhua Dong, and Chih-Jen Lin. 2020.  Unbiased Ad Click Prediction for Position-aware Advertising Systems. In _Proceedings of the 2020 ACM Conference on Recommender Systems_. ACM, 368–377.
  * Zhang et al. (2022b) An Zhang, Wenchang Ma, Xiang Wang, and Tat-Seng Chua. 2022b.  Incorporating Bias-aware Margins into Contrastive Loss for Collaborative Filtering.  _Proceedings of the 36th Conference on Neural Information Processing Systems_ (2022), 7866–7878.
  * Zhang et al. (2018) Brian Hu Zhang, Blake Lemoine, and Margaret Mitchell. 2018\.  Mitigating Unwanted Biases with Adversarial Learning. In _Proceedings of the 2018 AAAI/ACM Conference on AI, Ethics, and Society_. ACM, 335–340.
  * Zhang et al. (2021b) Dongyu Zhang, Minghao Zhang, Teng Guo, Ciyuan Peng, Vidya Saikrishna, and Feng Xia. 2021b.  In Your Face: Sentiment Analysis of Metaphor with Facial Expressive Features. In _Proceedings of the 35th Conference on Neural Information Processing Systems_. MIT Press, 1–8.
  * Zhang et al. (2022a) Tao Zhang, Jing Li, Mengde Han, Wanlei Zhou, Philip Yu, et al. 2022a.  Fairness in Semi-supervised Learning: Unlabeled Data Help to Reduce Discrimination.  _IEEE Transactions on Knowledge and Data Engineering_ 34, 4 (2022), 1763–1774.
  * Zhang and Ntoutsi (2019) Wenbin Zhang and Eirini Ntoutsi. 2019.  FAHT: An Adaptive Fairness-aware Decision Tree Classifier. In _Proceedings of the 2019 International Joint Conference on Artificial Intelligence_. Morgan Kaufmann.
  * Zhang et al. (2021a) Yang Zhang, Fuli Feng, Xiangnan He, Tianxin Wei, Chonggang Song, Guohui Ling, and Yongdong Zhang. 2021a.  Causal Intervention for Leveraging Popularity Bias in Recommendation. In _Proceedings of the 44th International ACM SIGIR Conference on Research and Development in Information Retreival_. ACM, 11–20.
  * Zhao et al. (2018) Jieyu Zhao, Yichao Zhou, Zeyu Li, Wei Wang, and Kai-Wei Chang. 2018.  Learning Gender-neutral Word Embeddings. In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_. ACL, 4847–4853.
  * Zhao et al. (2019) Kui Zhao, Junhao Hua, Ling Yan, Qi Zhang, Huan Xu, and Cheng Yang. 2019.  A Unified Framework for Marketing Budget Allocation. In _Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_. ACM, 1820–1830.
  * Zhao et al. (2022) Minghao Zhao, Le Wu, Yile Liang, Lei Chen, Jian Zhang, Qilin Deng, Kai Wang, Xudong Shen, Tangjie Lv, and Runze Wu. 2022\.  Investigating Accuracy-novelty Performance for Graph-based Collaborative Filtering. In _Proceedings of the 45th International ACM SIGIR Conference on Research and Development in Information Retrieval_. ACM, 50–59.
  * Zhao et al. (2023) Zihao Zhao, Jiawei Chen, Sheng Zhou, Xiangnan He, Xuezhi Cao, Fuzheng Zhang, and Wei Wu. 2023.  Popularity Bias is not Always Evil: Disentangling Benign and Harmful Bias for Recommendation.  _IEEE Transactions on Knowledge and Data Engineering_ 35, 10 (2023), 9920–9931.
  * Zhu et al. (2020) Ziwei Zhu, Jianling Wang, and James Caverlee. 2020.  Measuring and Mitigating Item Under-Recommendation Bias in Personalized Ranking Systems. In _Proceedings of the 2020 International ACM SIGIR Conference on Research and Development in Information Retrieval_. ACM, 449–458.
  * Zhuang et al. (2021) Honglei Zhuang, Zhen Qin, Xuanhui Wang, Mike Bendersky, Xinyu Qian, Po Hu, and Chary Chen. 2021.  Cross-positional Attention for Debiasing Clicks. ACM.
  * Zolaktaf et al. (2018) Zainab Zolaktaf, Reza Babanezhad, and Rachel Pottinger. 2018\.  A Generic Top-n Recommendation Framework for Trading-off Accuracy, Novelty, and Coverage. In _Proceedings of the 2018 IEEE International Conference on Data Engineering_. IEEE, 149–160.
  * Zou and Schiebinger (2018) James Zou and Londa Schiebinger. 2018.  AI can be Sexist and Racist—it’s Time to Make it Fair.  _Nature_ (2018), 324–326.


# On Certain Genus $0$ Entire Functions

Ruiming Zhang<EMAIL_ADDRESS>School of Mathematics and Computing
Sciences
Guilin University of Electronic Technology
Guilin, Guangxi 541004, P. R. China.

###### Abstract.

In this work we prove that an entire function $f(z)$ has only negative zeros
if and only if its order is strictly less $1$, its root sequence is real-part
dominating and for each nonnegative integer $k$ the real function
$\left(\frac{d}{dx}x\right)^{k}\left(\frac{f^{\prime}(x)}{f(x)}\right)$ is
completely monotonic on $(0,\infty)$. As an application we also state a
necessary and sufficient condition for the Riemann hypothesis.

###### Key words and phrases:

Laplace transforms; Complete monotonic functions; Transcendental special
functions.

###### 2000 Mathematics Subject Classification:

Primary 30C15; 44A10. Secondary 33C10; 11M26.

This work is supported by the National Natural Science Foundation of China,
grant No. 11771355.

##  1\.  Introduction

In this work we study the zeros of entire function $f(z)$,

(1.1) $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n},\quad a_{0}\cdot a_{n}>0,$

which is of order $\rho(f)$ strictly less than $1$ where $\rho(f)$ is defined
by

(1.2) $\rho(f)=\limsup_{n\to\infty}\frac{n\log n}{-\log\left|a_{n}\right|}.$

Many transcendental $q$-special functions such as Ramanujan’s entire function
$A_{q}(-z)$ is of this type, [7].

Since $\rho(f)<1$, then $f(z)$ is of genus $0$. If it has any roots, then
there exists a sequence of complex numbers
$\left\\{\lambda_{n}\right\\}_{n=1}^{\infty}$ such that [1, 3]

(1.3)
$\frac{f(z)}{f(0)}=\prod_{n=1}^{\infty}\left(1+\frac{z}{\lambda_{n}}\right)$

with

(1.4) $\sum_{n=1}^{\infty}\frac{1}{\left|\lambda_{n}\right|}<\infty.$

Furthermore, for any $1>\alpha_{0}>\rho(f)$, [3, Theorem 2.5.18]

(1.5)
$\sum_{n=1}^{\infty}\frac{1}{\left|\lambda_{n}\right|^{\alpha_{0}}}<\infty.$

More specifically, in this work we prove the following:

######  Theorem 1.

Let $f(z)$ be an entire function defined in (1.1), (1.3) such that it has a
nonzero root and it’s of order strictly less than $1$. If there is a positive
number $\beta_{0}$ with $\beta_{0}\in(0,1)$ such that the sequence
$\left\\{\lambda_{n}\right\\}_{n\in\mathbb{N}}$ in (1.3) and (1.4) satisfies

(1.6) $\Re(\lambda_{n})\geq\beta_{0}\left|\lambda_{n}\right|>0,\quad\forall
n\in\mathbb{N}.$

Then $f(z)$ has only negative zeros if and only if for each
$k\in\mathbb{N}_{0}$,

(1.7)
$G_{k}(x)=\left(\frac{d}{dx}x\right)^{k}\left(\frac{f^{\prime}(x)}{f(x)}\right),\quad\forall
x>0$

is completely monotonic on $(0,\infty)$.

Since the Ramanujan’s entire function $A_{q}(z)$ is an entire function of
order $0$ with only positive zeros for $0<q<1$, [7]. Then for all nonnegative
integer $k$ and $0<q<1$, as a consequence of Theorem 1, the function
$-\left(\frac{d}{dx}x\right)^{k}\left(\frac{A^{\prime}_{q}(-x)}{A_{q}(-x)}\right)$
is completely monotonic on $(0,\infty)$. However, many other transcendental
special functions are of the form

(1.8) $g(z)=\sum_{n=0}^{\infty}(-1)^{n}a_{n}z^{2n},\quad a_{0}\cdot a_{n}>0,$

for example, the Riemann $\Xi(z)$, and the Bessel functions
$\frac{J_{\nu}(z)}{z^{\nu}}$ , $\frac{J_{\nu}^{(2)}(z;q)}{z^{\nu}}$ with
$\nu>-1,\ 0<q<1$, [2, 7].

The order of $g(z)$ is strictly less than $2$,

(1.9)
$\rho(g)=\limsup_{n\to\infty}\frac{2n\log(2n)}{-\log\left|a_{n}\right|}=2\rho(f)<2.$

It is not hard to see that Theorem (1) can apply to $g(z)$ if it has at less
one zero and the condition (1.6) is replaced by

(1.10) $\Re(z_{n}^{2})>0,\quad\left|\Im(z_{n})\right|\leq M,\quad\forall
n\in\mathbb{N}.$

Here $M>0$ is a positive constant that only depends on $g(z)$, and
$\left\\{\pm z_{n}\right\\}_{n\in\mathbb{N}}$ are the roots of $g(z)$ with
$\Re(z_{n})>0,\ \forall n\in\mathbb{N}$.

Clearly, $f(z)=g(i\sqrt{z})$ transforms $g(z)$ in (1.8) into $f(z)$ in (1.1),
and the zeros $\left\\{\pm z_{n}\right\\}_{n\in\mathbb{N}}$ of $g(z)$ into
$\left\\{-\lambda_{n}\right\\}_{n\in\mathbb{N}}$, which are the zeros of
$f(z)$ with $\lambda_{n}=z_{n}^{2}$. Therefore, $\lambda_{n}>0$ if and only if
$z_{n}\in\mathbb{R}$.

Since $g(z)$ is an entire function not identically $0$, then $\left\\{\pm
z_{n}\right\\}_{n\in\mathbb{N}}$ has no limit points on the finite complex
plane. By (1.8) and (1.10) we must have

(1.11)
$\lim_{N\to\infty}\Re(z_{n})=+\infty,\quad\lim_{N\to\infty}\left|z_{n}\right|=+\infty.$

Without losing any generality we may let

(1.12)
$0<\Re(z_{1})\leq\Re(z_{2})<\dots\leq\Re(z_{n})\leq\Re(z_{n+1})\leq\dots.$

Then for any $\epsilon\in(0,1)$ there exists a $N_{\epsilon}\in\mathbb{N}$
such that

(1.13) $\epsilon\left|z_{n}\right|^{2}\geq 2M^{2},\quad\forall n\geq
N_{\epsilon}.$

Hence, for all $n\geq N_{\epsilon}$

(1.14)
$0<\Re(z_{n}^{2})=\left|z_{n}\right|^{2}-2\left(\Im(z_{n})\right)^{2}\geq\left|z_{n}\right|^{2}-2M^{2}\geq(1-\epsilon)\left|z_{n}\right|^{2}.$

Let

(1.15)
$\beta_{0}=\min\left\\{1-\epsilon,\frac{\Re(z_{n}^{2})}{\left|z_{n}^{2}\right|},\
1\leq n\leq N_{\epsilon}\right\\}\in(0,1),$

then it is clear that

(1.16) $\Re(z_{n}^{2})\geq\beta_{0}\left|z_{n}^{2}\right|,\quad\forall
n\in\mathbb{N},$

thus $\lambda_{n}=z_{n}^{2}$ satisfies (1.6). Therefore, if
$f(z)=g(i\sqrt{z})$ satisfies (1.7) additionally, then by Theorem 1 $f(z)$ has
only negative zeros, which is equivalent to that $g(z)$ has only real zeros.

It is known that for $\nu>-1$ the even entire function
$\frac{J_{\nu}(z)}{z^{\nu}}$ is of order $1$ with only real zeros, while for
$\nu>-1,\ 0<q<1$ the $q$-Bessel functions $\frac{J_{\nu}^{(2)}(z;q)}{z^{\nu}}$
are even entire functions that of order $0$ with only real zeros, [7].
Applying Theorem 1 to the Bessel functions we conclude that for each
nonnegative integer $k$ and a positive number $q$ with $0<q<1$, the functions
$\left(\frac{d}{dx}x\right)^{k}\left(\frac{f^{\prime}(x)}{f(x)}\right)$ are
completely monotonic on $(0,\infty)$ where $f(x)=x^{-\nu/2}I_{\nu}(\sqrt{x})$
or $f(x)=x^{-\nu/2}I_{\nu}^{(2)}(\sqrt{x};q)$.

##  2\. Proof of the Theorem

###  2.1. the heat kernel $\Theta(x)$

######  Lemma 2.

Let $f(z)$ be an entire function defined in (1.1), (1.3) with order
$\rho(f)<1$ such that there exists a positive number $\beta_{0}\in(0,1)$ it
satisfies (1.6). Then its associated heat kernel

(2.1) $\Theta(t)=\sum_{n=1}^{\infty}e^{-\lambda_{n}t},\quad t>0$

is in $C^{\infty}(0,\infty)$, and for any

(2.2) $k\in\mathbb{N}_{0},\ 1>\alpha>p(f),\
0<\beta<\beta_{0}\inf\left\\{\left|\lambda_{n}\right|\bigg{|}n\in\mathbb{N}\right\\},$

the function $\Theta(t)$ satisfies

(2.3) $\Theta^{(k)}(t)=\mathcal{O}\left(t^{-\alpha-k}\right),\quad t\to 0^{+}$

and

(2.4) $\Theta^{(k)}(t)=\mathcal{O}\left(e^{-\beta t}\right),\quad
t\to+\infty.$

Furthermore, for all $k\in\mathbb{N}_{0},x\geq 0$,

(2.5)
$\int_{0}^{\infty}t^{k}e^{-xt}\left|\Theta(t)\right|dt\leq\frac{k!}{\beta_{0}^{k+1}}\sum_{n=1}^{\infty}\frac{1}{\left|\lambda_{n}\right|^{k+1}}<\infty$

and

(2.6)
$(-1)^{k}\left(\frac{f^{\prime}(x)}{f(x)}\right)^{(k)}=\sum_{n=1}^{\infty}\frac{k!}{(x+\lambda_{n})^{k+1}}=\int_{0}^{\infty}e^{-xt}t^{k}\Theta(t)dt.$

###### Proof.

Since for $a>0$

(2.7) $\sup_{x>0}x^{a}e^{-x}=\left(\frac{a}{e}\right)^{a},$

then for any $t>0,\,k\geq 0$ by (1.5) and (1.6),

(2.8)
$\displaystyle\sum_{n=1}^{\infty}\left|\lambda_{n}^{k}e^{-\lambda_{n}t}\right|=\sum_{n=1}^{\infty}\left|\lambda_{n}\right|^{k}e^{-\Re(\lambda_{n})t}\leq\frac{1}{(\beta_{0}t)^{\alpha_{0}+k}}\sum_{n=1}^{\infty}\frac{\left(\beta_{0}t\left|\lambda_{n}\right|\right)^{k+\alpha_{0}}e^{-\beta_{0}\left|\lambda_{n}\right|t}}{\left|\lambda_{n}\right|^{\alpha_{0}}}$
$\displaystyle\leq\frac{\sup_{x>0}x^{k+\alpha_{0}}e^{-x}}{(\beta_{0}t)^{\alpha_{0}+k}}\sum_{n=1}^{\infty}\frac{1}{\left|\lambda_{n}\right|^{\alpha_{0}}}=\left(\frac{k+\alpha_{0}}{e\beta_{0}t}\right)^{k+\alpha_{0}}\sum_{n=1}^{\infty}\frac{1}{\left|\lambda_{n}\right|^{\alpha_{0}}}<\infty,$

which proves that $\Theta(t)\in C^{\infty}(0,\infty)$ and (2.3).

Let
$0<\beta<\beta_{0}\inf\left\\{\left|\lambda_{n}\right|\bigg{|}n\in\mathbb{N}\right\\}$,
then there exists a positive number $\epsilon$ with $0<\epsilon<1$ such that

(2.9)
$\beta=\epsilon\beta_{0}\inf\left\\{\left|\lambda_{n}\right|\bigg{|}n\in\mathbb{N}\right\\}\leq\epsilon\beta_{0}\left|\lambda_{n}\right|,\quad\forall
n\in\mathbb{N}.$

Thus for $t\geq 1$ by (2.8),

(2.10) $\displaystyle\left|e^{\beta
t}\Theta^{(k)}(t)\right|\leq\sum_{n=1}^{\infty}\left|\lambda_{n}\right|^{k}e^{-(\beta_{0}\left|\lambda_{n}\right|-\beta)t}\leq\sum_{n=1}^{\infty}\left|\lambda_{n}\right|^{k}e^{-(1-\epsilon)\beta_{0}\left|\lambda_{n}\right|t}$
$\displaystyle\leq\sum_{n=1}^{\infty}\left|\lambda_{n}\right|^{k}e^{-(1-\epsilon)\beta_{0}\left|\lambda_{n}\right|}<\infty,$

which establishes (2.4).

For all $x,k\geq 0$ since

(2.11)
$\displaystyle\int_{0}^{\infty}t^{k}e^{-xt}\left|\Theta(t)\right|dt\leq\int_{0}^{\infty}t^{k}\left(\sum_{n=1}^{\infty}e^{-(x+\Re(\lambda_{n}))t}\right)dt\leq\sum_{n=1}^{\infty}\int_{0}^{\infty}t^{k}e^{-(x+\beta_{0}\left|\lambda_{n}\right|)t}dt$
$\displaystyle\leq\sum_{n=1}^{\infty}\frac{k!}{\left(x+\beta_{0}\left|\lambda_{n}\right|\right)^{k+1}}\leq\frac{k!}{\beta_{0}^{k+1}}\sum_{n=1}^{\infty}\frac{1}{\left|\lambda_{n}\right|^{k+1}}<\infty,$

then,

(2.12)
$\displaystyle\left(\frac{f^{\prime}(x)}{f(x)}\right)^{(k)}=\sum_{n=1}^{\infty}\frac{(-1)^{k}k!}{(x+\lambda_{n})^{k+1}}$
$\displaystyle=(-1)^{k}\sum_{n=1}^{\infty}\int_{0}^{\infty}t^{k}e^{-(x+\lambda_{n})t}dt=(-1)^{k}\int_{0}^{\infty}e^{-xt}t^{k}\Theta(t)dt.$

∎

######  Lemma 3.

Let $f(z)$, $\Theta(x)$ and $\left\\{\lambda_{n}\right\\}_{n=1}^{\infty}$ be
defined as in Theorem 1 and Lemma 2. Then the sequence
$\left\\{\lambda_{n}\right\\}_{n=1}^{\infty}$ is positive if and only if the
function $\Theta(x)$ is completely monotonic on $(0,\infty)$.

###### Proof.

Assume that $\left\\{\lambda_{n}\right\\}_{n=1}^{\infty}$ is positive, then

(2.13)
$(-1)^{k}\Theta^{(k)}(t)=\sum_{n=1}^{\infty}\lambda_{n}^{k}e^{-\lambda_{n}t}\geq
0,\quad\forall t>0,k\in\mathbb{N}_{0},$

hence $\Theta(t)$ is completely monotonic by definition.

On the other hand if $\Theta(t)$ is completely monotonic on $(0,\infty)$, then
by the Hausdorff–Bernstein–Widder theorem there exists a nonnegative measure
$d\mu(t)$ on $[0,\infty)$ such that [8, 9]

(2.14)
$\Theta(t)=\sum_{n=1}^{\infty}e^{-\lambda_{n}t}=\int_{0}^{\infty}e^{-yt}d\mu(y),\quad\forall
t>0.$

For any $x\geq 0$ we multiply $e^{-xt}$ both sides of (2.14) and integrate it
with respect to $t$ from $0$ to $\infty$ to get

(2.15)
$\displaystyle\int_{0}^{\infty}e^{-xt}\Theta(t)dt=\sum_{n=1}^{\infty}\frac{1}{x+\lambda_{n}}=\frac{f^{\prime}(x)}{f(x)}$
$\displaystyle=\int_{0}^{\infty}e^{-xt}\left(\int_{0}^{\infty}e^{-yt}d\mu(y)\right)dx=\int_{0}^{\infty}\frac{d\mu(y)}{x+y}.$

The exchange of orders between summation and integration, and between
integrations are guaranteed by (2.5), (2.6) and Fubini’s theorem.

By Fatou’s Lemma and (2.5), (2.6) and (2.15),

(2.16) $\displaystyle
0<\int_{0}^{\infty}\frac{d\mu(y)}{y}=\int_{0}^{\infty}\liminf_{x\downarrow
0}\frac{d\mu(y)}{x+y}\leq\liminf_{x\downarrow
0}\int_{0}^{\infty}\frac{d\mu(y)}{x+y}$ $\displaystyle=\liminf_{x\downarrow
0}\left|\frac{f^{\prime}(x)}{f(x)}\right|\leq\liminf_{x\downarrow
0}\left|\sum_{n=1}^{\infty}\frac{1}{x+\Re(\lambda_{n})}\right|\leq\frac{1}{\beta_{0}}\sum_{n=1}^{\infty}\frac{1}{\left|\lambda_{n}\right|}.$

For $x,y>0$ since $\frac{1}{(x+y)^{2}}<\frac{1}{4xy}$ then,

(2.17)
$\int_{0}^{\infty}\frac{d\mu(y)}{(x+y)^{2}}\leq\frac{1}{4x}\int_{0}^{\infty}\frac{d\mu(y)}{y}<\infty.$

For $x,y>0,\ \left|h\right|<\frac{x}{2}$ by (2.15), Lebesgue’s dominated
convergence theorem and the inequality

(2.18)
$\frac{2}{\left|(x+y)(x+y+h)\right|}\leq\frac{1}{(x+y)^{2}}+\frac{1}{(x/2+y)^{2}}$

we obtain

(2.19)
$\displaystyle-\frac{d}{dx}\left(\frac{f^{\prime}(x)}{f(x)}\right)=-\frac{d}{dx}\int_{0}^{\infty}\frac{d\mu(y)}{x+y}$
$\displaystyle=\lim_{h\to
0}\frac{1}{h}\int_{0}^{\infty}\left(\frac{1}{x+y}-\frac{1}{x+y+h}\right)d\mu(y)$
$\displaystyle=\lim_{h\to
0}\int_{0}^{\infty}\frac{d\mu(y)}{(x+y)(x+y+h)}=\int_{0}^{\infty}\frac{d\mu(y)}{(x+y)^{2}}.$

Then by Fatou’s Lemma and (2.6),

(2.20) $\displaystyle
0<\int_{0}^{\infty}\frac{d\mu(y)}{y^{2}}=\int_{0}^{\infty}\liminf_{x\downarrow
0}\frac{d\mu(y)}{(x+y)^{2}}\leq\liminf_{x\downarrow
0}\int_{0}^{\infty}\frac{d\mu(y)}{(x+y)^{2}}$
$\displaystyle=\liminf_{x\downarrow
0}\left|\frac{d}{dx}\left(\frac{f^{\prime}(x)}{f(x)}\right)\right|\leq\liminf_{x\downarrow
0}\left|\sum_{n=1}^{\infty}\frac{1}{(x+\Re(\lambda_{n}))^{2}}\right|\leq\frac{1}{\beta_{0}^{2}}\sum_{n=1}^{\infty}\frac{1}{\left|\lambda_{n}\right|^{2}}.$

Let

(2.21) $s(z)=\int_{0}^{\infty}\frac{d\mu(y)}{z+y},$

we first show that $s(z)$ is continuous and bounded on $\Re(z)\geq 0$ and
analytic in $\Re(z)>0$.

For $\Re(z)\geq 0$ since

(2.22)
$\int_{0}^{\infty}\frac{d\mu(y)}{\left|z+y\right|}\leq\int_{0}^{\infty}\frac{d\mu(y)}{\Re(z)+y}\leq\int_{0}^{\infty}\frac{d\mu(y)}{y}<\infty$

and

(2.23)
$\int_{0}^{\infty}\frac{d\mu(y)}{\left|z+y\right|^{2}}\leq\int_{0}^{\infty}\frac{d\mu(y)}{(\Re(z)+y)^{2}}\leq\int_{0}^{\infty}\frac{d\mu(y)}{y^{2}}<\infty,$

then both integrals

(2.24)
$\int_{0}^{\infty}\frac{d\mu(y)}{z+y},\quad\int_{0}^{\infty}\frac{d\mu(y)}{(z+y)^{2}}$

converge absolutely for $\Re(z)\geq 0$ and uniformly on any of its compact
subsets. Thus they are continuous on $\Re(z)\geq 0$ by (2.22) and (2.23).
Similar to the proof of (2.19) we can show that

(2.25)
$s^{\prime}(z)=-\int_{0}^{\infty}\frac{d\mu(y)}{(z+y)^{2}},\quad\text{$\Re(z)>0,$
}$

Then $s(z)$ is analytic in $\Re(z)>0$.

Next we prove that $s(z)$ is continuous on any of its compact subsets of
$\left\\{z|\Re(z)\leq 0,\,\Im(z)\neq 0\right\\}$ and analytic in
$\left\\{z|\Re(z)<0,\,\Im(z)\neq 0\right\\}$.

Let $\Re(z)=-\alpha\leq 0$ and $\beta=\Im(z)\neq 0$, then for any positive
number $\epsilon>0$,

(2.26) $\displaystyle\int_{0}^{\infty}\frac{d\mu(y)}{\left|(y-\alpha)+\beta
i\right|}=\int_{0}^{\alpha+\epsilon}\frac{d\mu(y)}{\left|(y-\alpha)+\beta
i\right|}+\int_{\alpha+\epsilon}^{\infty}\frac{d\mu(y)}{\left|(y-\alpha)+\beta
i\right|}$
$\displaystyle\leq\frac{1}{\left|\beta\right|}\int_{0}^{\alpha+\epsilon}\frac{yd\mu(y)}{y}+\int_{\alpha+\epsilon}^{\infty}\frac{y}{(y-\alpha)}\frac{d\mu(y)}{y}$
$\displaystyle\leq\frac{\alpha+\epsilon}{\left|\beta\right|}\int_{0}^{\alpha+\epsilon}\frac{d\mu(y)}{y}+\int_{\alpha+\epsilon}^{\infty}\frac{y-\alpha+\alpha}{y-\alpha}\frac{d\mu(y)}{y}$
$\displaystyle\leq\frac{\alpha+\epsilon}{\left|\beta\right|}\int_{0}^{\alpha+\epsilon}\frac{d\mu(y)}{y}+\left(1+\frac{\alpha}{\epsilon}\right)\int_{\alpha+\epsilon}^{\infty}\frac{d\mu(y)}{y}$
$\displaystyle\leq\left(1+\frac{\alpha}{\epsilon}+\frac{\alpha+\epsilon}{\left|\beta\right|}\right)\int_{0}^{\infty}\frac{d\mu(y)}{y}<\infty$

and

(2.27) $\displaystyle\int_{0}^{\infty}\frac{d\mu(y)}{\left|(y-\alpha)+\beta
i\right|^{2}}\leq\frac{1}{\beta^{2}}\int_{0}^{\alpha+\epsilon}\frac{y^{2}d\mu(y)}{y^{2}}+\int_{\alpha+\epsilon}^{\infty}\frac{(y-\alpha+\alpha)^{2}}{(y-\alpha)^{2}}\frac{d\mu(y)}{y^{2}}$
$\displaystyle\leq\left(\frac{\alpha+\epsilon}{\beta}\right)^{2}\int_{0}^{\alpha+\epsilon}\frac{d\mu(y)}{y^{2}}+\int_{\alpha+\epsilon}^{\infty}\left(1+\frac{\alpha}{y-\alpha}\right)^{2}\frac{d\mu(y)}{y^{2}}$
$\displaystyle\leq\left(\frac{\alpha+\epsilon}{\beta}\right)^{2}\int_{0}^{\alpha+\epsilon}\frac{d\mu(y)}{y^{2}}+\left(1+\frac{\alpha}{\epsilon}\right)^{2}\int_{\alpha+\epsilon}^{\infty}\frac{d\mu(y)}{y^{2}}$
$\displaystyle\leq\left(\left(\frac{\alpha+\epsilon}{\beta}\right)^{2}+\left(1+\frac{\alpha}{\epsilon}\right)^{2}\right)\int_{0}^{\infty}\frac{d\mu(y)}{y^{2}}<\infty.$

Then both integrals

(2.28)
$\int_{0}^{\infty}\frac{d\mu(y)}{y+z},\quad\int_{0}^{\infty}\frac{d\mu(y)}{\left(y+z\right)^{2}}$

converge absolutely and uniformly on any of the compact subsets of
$\left\\{z|\Re(z)\leq 0,\,\Im(z)\neq 0\right\\}$. Similar to the case
$\Re(z)\geq 0$, the above estimates imply that $s(z)$ is continuous on
$\left\\{z|\Re(z)\leq 0,\,\Im(z)\neq 0\right\\}$, analytic in
$\left\\{z|\Re(z)<0,\,\Im(z)\neq 0\right\\}$.

For any $z_{0}$ with $\Re(z_{0})=0$ and $\Im(z_{0})\neq 0$, there exists a
positive number $\epsilon>0$ such that the disk
$D(z_{0},\epsilon)=\left\\{z:\left|z-z_{0}\right|<\epsilon\right\\}$ is
completely contained in
$\left\\{z:z\in\mathbb{C}\backslash\left(-\infty,0\right],\,\Im(z)>0\right\\}$
or
$\left\\{z:z\in\mathbb{C}\backslash\left(-\infty,0\right],\,\Im(z)<0\right\\}$.
Without losing any generality we may assume that $\Im(z_{0})>0$ and
$D(z_{0},\epsilon)\subset\left\\{z:z\in\mathbb{C}\backslash\left(-\infty,0\right],\,\Im(z)>0\right\\}$,
and the other case can be treated similarly.

We first observe that $s(z)$ is continuous on the closure of
$D(z_{0},\epsilon/3)$ except possibly on its intersection with the imaginary
axis $\Re(z)=0$ where it has only one-sided continuity. For any $z$ in the
intersection, let $\left\\{z_{j}\right\\}$ be a sequence in the closure of
$D(z_{0},\epsilon/3)$ that converges to $z$. If all $\left\\{z_{j}\right\\}$
except finitely many are completely inside one of $\Re(z)\geq 0$ and
$\left\\{z|\Re(z)\leq 0,\,\Im(z)\geq\epsilon/3\right\\}$, then by the one-
sided continuity we have $s(z_{j})\to s(z)$ as $j\to\infty$. Otherwise, the
sequence can be decomposed into two converging subsequences
$\left\\{z_{j_{1}}\right\\}$ and $\left\\{z_{j_{2}}\right\\}$, which are
completely contained inside in $\Re(z)\geq 0$ and $\left\\{z|\Re(z)\leq
0,\,\Im(z)\geq\epsilon/3\right\\}$ respectively. However, by the one-sided
continuities proved on both sets we have $s(z_{j_{k}})\to s(z)$ as
$j_{k}\to\infty$ for $k=1,2$. Therefore, $s(z)$ is continuous on the closure
of $D(z_{0},\epsilon/3)$.

Given any closed piecewise $C^{1}$ curve $\gamma$ in $D(z_{0},\epsilon/3)$. If
it does not intersect the imaginary axis $\Re(z)=0$, then it’s completely
contained in one of two disjoint regions, $\Re(z)>0$ and
$\left\\{z|\Re(z)<0,\,\Im(z)>0\right\\}$. From the above discussion $s(z)$ is
analytic in both of them, then by Cauchy’s integration theorem we have
$\oint_{\gamma}s(z)dz=0$. Otherwise, $\gamma$ can be decomposed into a sum of
closed piecewise $C^{1}$ curves $\left\\{\gamma_{j}\right\\}$ in
$D(z_{0},\epsilon/3)$ by adding the line segments with opposed directions on
$\Re(z)=0$ enclosed by $\gamma$. Then $s(z)$ is continuous on the closed sets
enclosed by each $\gamma_{j}$ and analytic in their interiors. By a weaker
version of Cauchy’s integration theorem for each $j$ we have
$\oint_{\gamma_{j}}s(z)dz=0$ and [1]

(2.29) $\oint_{\gamma}s(z)dz=\sum_{j}\oint_{\gamma_{j}}s(z)dz=\sum_{j}0=0.$

Then by Morera’s theorem $s(z)$ is analytic in $D(z_{0},\epsilon/3)$, in
particular at $z_{0}$.

In the summary of above discussion we have proved that $s(z)$ is continuous on
$\mathbb{C}\backslash\left(-\infty,0\right)$ and analytic in
$\mathbb{C}\backslash\left(-\infty,0\right]$.

Since the meromorphic function

(2.30) $\frac{f^{\prime}(z)}{f(z)}=\sum_{n=1}^{\infty}\frac{1}{z+\lambda_{n}}$

is analytic in
$\mathbb{C}\backslash\left\\{-\lambda_{n}\right\\}_{n\in\mathbb{N}}$, and
$s(z)$ is analytic in $\mathbb{C}\backslash\left(-\infty,0\right]$, while by
(2.15) they are equal on $z\geq 0$, then we must have

(2.31) $\frac{f^{\prime}(z)}{f(z)}=s(z),\quad
z\in\mathbb{C}\backslash\left(\left\\{-\lambda_{n}\right\\}_{n\in\mathbb{N}}\cup\left(-\infty,0\right]\right)$

by analytic continuation.

If there exists an $n_{0}\in\mathbb{N}$ such that $\lambda_{n_{0}}$ is not
positive. Since $\Re(\lambda_{n_{0}})>0$ and $-\lambda_{n_{0}}$ is not
negative, then $s(z)$ is analytic at $-\lambda_{n_{0}}$. Hence,

(2.32)
$\lim_{z\to-\lambda_{n_{0}}}s(z)=s\left(-\lambda_{n_{0}}\right)=\int_{0}^{\infty}\frac{d\mu(y)}{y-\lambda_{n_{0}}}\neq\infty.$

On the other hand since $-\lambda_{n_{0}}$ is a zero of $f(z)$, then it is a
simple pole of $\frac{f^{\prime}(z)}{f(z)}$ with a positive residue. Hence,

(2.33) $\lim_{z\to-\lambda_{n_{0}}}\frac{f^{\prime}(z)}{f(z)}=\infty.$

Combining (2.32) and (2.33) we have reached the following contradiction,

(2.34) $\infty\neq
s\left(-\lambda_{n_{0}}\right)=\lim_{z\to-\lambda_{n_{0}}}s(z)=\lim_{z\to-\lambda_{n_{0}}}\frac{f^{\prime}(z)}{f(z)}=\infty.$

This contradiction proves that $\left\\{\lambda_{n}\right\\}_{n\in\mathbb{N}}$
must be a positive sequence. ∎

###  2.2. Proof of Theorem

For $k\in\mathbb{N}_{0},\,t>0$ and $z$ with $\Re(z)>0$ we define

(2.35) $\Theta_{k}(t)=(-t)^{k}\Theta^{(k)}(t),\quad
G_{k}(z)=\int_{0}^{\infty}e^{-zt}\Theta_{k}(t)dt.$

For any $k\in\mathbb{N}_{0}$ and $t>0$, it is clear that
$(-1)^{k}\Theta^{(k)}(t)\geq 0$ if and only if $\Theta_{k}(t)\geq 0$. By
(2.3), (2.4) and (2.6) $G_{k}(x)$ exists for all $x,k\geq 0$ and is in
$C^{\infty}(0,\infty)$.

For each $k\in\mathbb{N}_{0}$, apply the Hausdorff–Bernstein–Widder theorem to
$G_{k}(x)$ we get that $\Theta_{k}(t)\geq 0$ if and only if $G_{k}(x)$ is
completely monotonic on $(0,\infty)$.

For any $k\in\mathbb{N}$ and $x>0$, by (2.3) and (2.4) again,

(2.36) $\displaystyle
G_{k}(x)=\int_{0}^{\infty}e^{-xt}\Theta_{k}(t)dt=-\int_{0}^{\infty}te^{-xt}\Theta^{\prime}_{k-1}(t)dt$
$\displaystyle=\int_{0}^{\infty}\left(1-xt\right)e^{-xt}\Theta_{k-1}(t)dt=\frac{d}{dx}\left(xG_{k-1}(x)\right)$
$\displaystyle=\left(\frac{d}{dx}x\right)^{k}G_{0}(x)=\left(\frac{d}{dx}x\right)^{k}\left(\frac{f^{\prime}(x)}{f(x)}\right).$

##  3\. An Application

The Riemann $\xi$-function is defined by [2, 4, 5, 6],

(3.1) $\xi(s)=\pi^{-s/2}(s-1)\Gamma\left(1+\frac{s}{2}\right)\zeta(s),\quad
s=\sigma+it,\ \sigma,t\in\mathbb{R},$

where $\Gamma(s)$, $\zeta(s)$ are the respective analytic continuations of

(3.2) $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}dx,\quad\sigma>0$

and

(3.3) $\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}},\quad\sigma>1.$

It is well-known that $\xi(s)$ is an order $1$ entire function that satisfies
$\xi(s)=\xi(1-s)$, and it has infinitely many zeros, all of them are in the
vertical strip $\sigma\in(0,1)$. Then the Riemann Xi function
$\Xi(s)=\xi\left(\frac{1}{2}+is\right)$ is an even entire function of order
$1$, and it has infinitely many zeros, all of them are in the horizontal strip
$t\in(-1/2,1/2)$. The Riemann hypothesis is that all the zeros of $\xi(s)$ are
on $\sigma=\frac{1}{2},$or equivalently, all the zeros of $\Xi(s)$ are real.

Since [4, 5, 6]

(3.4)
$\Xi(s)=\int_{-\infty}^{\infty}\Phi(u)e^{ius}du=2\int_{0}^{\infty}\Phi(u)\cos(us)du,$

where

(3.5)
$\Phi(u)=\Phi(-u)=\sum_{n=1}^{\infty}\left(4n^{4}\pi^{2}e^{9u/2}-6n^{2}\pi
e^{5u/2}\right)e^{-n^{2}\pi e^{2u}}>0.$

Then,

(3.6) $\xi\left(\frac{1}{2}+s\right)=\sum_{n=0}^{\infty}a_{n}s^{2n},\quad
a_{n}=\frac{2}{(2n)!}\int_{0}^{\infty}\Phi(u)u^{2n}du>0$

and

(3.7)
$f(s)=\xi\left(\frac{1}{2}+\sqrt{s}\right)=\sum_{n=0}^{\infty}a_{n}s^{n}$

defines an entire function of order $\frac{1}{2}$, and $\Xi(s)$ has only real
roots if and only if $f(s)$ has only negative roots. Apply Theorem 1 to $f(s)$
we obtain the following corollary:

######  Corollary 4.

The Riemann hypothesis is true if and only if for each nonnegative integer
$k$, the function

(3.8)
$\left(\frac{d}{dx}x\right)^{k}\left(\frac{\xi^{\prime}\left(\frac{1}{2}+\sqrt{x}\right)}{\sqrt{x}\xi\left(\frac{1}{2}+\sqrt{x}\right)}\right)$

is completely monotonic on $(0,\infty)$.

## References

  * [1] L. Ahlfors, _Complex Analysis_ , 3rd edition, McGraw-Hill Education, 1979.
  * [2] G. Andrews, R. Askey and R. Roy, _Special Functions_ , 1st edition, Cambridge University Press.
  * [3] R. P. Boas, _Entire Functions_ , 1st edition, Academic Press, 1954.
  * [4] H. M. Edwards, _Riemann’s Zeta Function_ , Dover Publications, 2001.
  * [5] George Gasper, Using sums of squares to prove that certain entire functions have only real zeros, dedicated to the memory of Ralph P. Boas, Jr. (1912–1992), published in _Fourier Analysis: Analytic and Geometric Aspects_ , W.O. Bray, P.S. Milojevic and C.V. Stanojevic, eds., Marcel Dekker, 1994, pp. 171–186.
  * [6] G. Gasper, Using integrals of squares to prove that certain real-valued special functions to prove that the Pólya $\Xi^{*}\left(z\right)$ function, the functions $K_{iz}\left(a\right),\ a>0$ and some other entire functions have only real zeros, dedicated to Dan Waterman on the occasion of his 80th birthday.
  * [7] M. E. H. Ismail, _Classical and Quantum Orthogonal Polynomials in One Variable,_ Cambridge University Press, Cambridge, 2005\.
  * [8] R. Schilling, R. Song and Z. Vondracek, _Bernstein Functions: Theory and Applications_ , 1st edition, de Gruyter, 2010.
  * [9] D. V. Widder, _The Laplace Transform_ , Princeton Mathematical Series, v. 6, Princeton University Press, 1941.


# Restrained Italian domination in trees

Kijung Kim  Department of Mathematics, Pusan National University, Busan 46241,
Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

Let $G=(V,E)$ be a graph. A subset $D$ of $V$ is a restrained dominating set
if every vertex in $V\setminus D$ is adjacent to a vertex in $D$ and to a
vertex in $V\setminus D$. The restrained domination number, denoted by
$\gamma_{r}(G)$, is the smallest cardinality of a restrained dominating set of
$G$. A function $f:V\rightarrow\\{0,1,2\\}$ is a restrained Italian dominating
function on $G$ if (i) for each vertex $v\in V$ for which $f(v)=0$, it holds
that $\sum_{u\in N_{G}(v)}f(u)\geq 2$, (ii) the subgraph induced by $\\{v\in
V\mid f(v)=0\\}$ has no isolated vertices. The restrained Italian domination
number, denoted by $\gamma_{rI}(G)$, is the minimum weight taken over all
restrained Italian dominating functions of $G$. It is known that
$\gamma_{r}(G)\leq\gamma_{rI}(G)\leq 2\gamma_{r}(G)$ for any graph $G$. In
this paper, we characterize the trees $T$ for which
$\gamma_{r}(T)=\gamma_{rI}(T)$, and we also characterize the trees $T$ for
which $\gamma_{rI}(T)=2\gamma_{r}(T)$.

Key words: restrained domination, restrained Italian domination, tree

###### 2010 Mathematics Subject Classification:

05C69

This research was supported by Basic Science Research Program through the
National Research Foundation of Korea funded by the Ministry of Education
(2020R1I1A1A01055403).

##  1\. Introduction and Terminology

Let $G=(V,E)$ be a finite simple graph with vertex set $V=V(G)$ and edge set
$E=E(G)$. The open neighborhood of $v\in V(G)$ is the set $N_{G}(v)=\\{u\in
V(G)\mid uv\in E(G)\\}$ and the closed neighborhood of $v\in V(G)$ is the set
$N_{G}[v]:=N_{G}(v)\cup\\{v\\}$. A subset $D$ of $V(G)$ is a dominating set if
every vertex in $V(G)\setminus D$ is adjacent to a vertex in $D$. The
domination number of $G$, denoted by $\gamma(G)$, is the minimum cardinality
of a dominating set in $G$. A dominating set with the cardinality $\gamma(G)$
is called a $\gamma(G)$-set.

In [3], Domke et al. gave the formal definition of restrained domination. A
subset $S$ of $V(G)$ is a restrained dominating set (RDS) if every vertex in
$V(G)\setminus S$ is adjacent to a vertex in $S$ and another vertex in
$V(G)\setminus S$. The restrained domination number of $G$, denoted by
$\gamma_{r}(G)$, is the minimum cardinality of a restrained dominating set in
$G$. A restrained dominating set with the cardinality $\gamma_{r}(G)$ is
called a $\gamma_{r}(G)$-set. As explained in [3], there is one possible
application of the concept of restrained domination. Each vertex in a RDS $S$
represents a guard and each vertex in $V(G)\setminus S$ represents a prisoner.
Each prisoner must be observed by at least one guard and every prisoner must
be seen by at least one other prisoner to protect the rights of prisoners. To
be cost effective, it is desirable to place as few guards as possible.

A function $f:V(G)\rightarrow\\{0,1,2\\}$ is an Italian dominating function on
$G$ if for each vertex $v\in V(G)$ for which $f(v)=0$, it holds that
$\sum_{u\in N_{G}(v)}f(u)\geq 2$. In [6], Samadi et al. introduced the concept
of restrained Italian domination as a variant of Italian dominating function.
An Italian dominating function $f:V\rightarrow\\{0,1,2\\}$ is a restrained
Italian dominating function (RIDF) on $G$ if the subgraph induced by $\\{v\in
V\mid f(v)=0\\}$ has no isolated vertices. A RIDF $f$ gives an ordered
partition $(V_{0},V_{1},V_{2})$ (or $(V_{0}^{f},V_{1}^{f},V_{2}^{f})$ to refer
to $f$) of $V(G)$, where $V_{i}:=\\{v\in V(G)\mid f(v)=i\\}$. The weight of a
RIDF $f$ is $\omega(f):=\sum_{v\in V}f(v)$. The restrained Italian domination
number, denoted by $\gamma_{rI}(G)$, is the minimum weight taken over all
restrained Italian dominating functions of $G$. A $\gamma_{rI}(G)$-function is
a RIDF on $G$ with weight $\gamma_{rI}(G)$.

As noted in [6, Proposition 3.3], it holds that
$\gamma_{r}(G)\leq\gamma_{rI}(G)\leq 2\gamma_{r}(G)$ for any graph $G$. We
define a tree $T$ to be a $(\gamma_{r},\gamma_{rI})$-tree if
$\gamma_{r}(T)=\gamma_{rI}(T)$. We define a tree $T$ to be a restrained
Italian tree if $\gamma_{rI}(T)=2\gamma_{r}(T)$. In this paper, we
characterize $(\gamma_{r},\gamma_{rI})$-trees and restrained Italian trees.

The rest of this section, we present some necessary terminology and notation.
For terminology and notation on graph theory not given here, the reader is
referred to [1, 7]. The degree of $v\in V(G)$ is defined as the cardinality of
$N_{G}(v)$, denoted by $deg_{G}(v)$. A diametral path of $G$ is a path with
the length which equals the diameter of $G$. A subset $S$ of $V(G)$ is a
packing in $G$ if the vertices of $S$ are pairwise at distance at least three
apart in $G$. The packing number of $G$, denoted by $\rho(G)$, is the maximum
cardinality of a packing in $G$. A packing with the cardinality $\rho(G)$ is
called a $\rho(G)$-set.

Let $T$ be a (rooted) tree. A leaf of $T$ is a vertex of degree one. A stem
(or support vertex) is a vertex adjacent to a leaf. A weak stem is a stem that
is adjacent to exactly one leaf. For a vertex $v$ in a rooted tree, we let
$C(v)$ and $D(v)$ denote the set of children and descendants, respectively, of
$v$. The subtree induced by $D(v)\cup\\{v\\}$ is denoted by $T_{v}$. We write
$K_{1,n-1}$ for the star of order $n\geq 3$. The double star $DS_{p,q}$, where
$p,q\geq 1$, is the graph obtained by joining the centers of two stars
$K_{1,p}$ and $K_{1,q}$. A healthy spider $S_{t,t}$ is the graph from a star
$K_{1,t}$ by subdividing each edges of $K_{1,t}$. For two graph $G$ and $H$,
if $G$ is isomorphic to $H$, we denote it by $G\cong H$. For a graph $G$ and
its subgraph $S$, $G-S$ denotes the subgraph of $G$ induced by $V(G)\setminus
V(S)$.

##  2\. $(\gamma_{r},\gamma_{rI})$-trees

In this section, we characterize the trees for which
$\gamma_{rI}(T)=\gamma_{r}(T)$. First, we introduce a family $\mathcal{H}$ of
trees that can be obtained from a sequence $T_{1},T_{2},\dotsc,T_{m}$ $(m\geq
1)$ of trees such that $T_{1}$ is a double star $DS_{l,n}$ $(l,n\geq 2)$, and
if $m\geq 2$, $T_{i+1}$ can be obtained recursively from $T_{i}$ by one of the
following operations for $1\leq i\leq m-1$.

Define

$LV(T_{i})=\\{v\in V(T_{i})\mid v~{}\text{is a leaf of}~{}T_{j}~{}\text{for
some}~{}j\leq i\\}$

and

$SV(T_{i})=\\{v\in V(T_{i})\mid v~{}\text{is a stem of}~{}T_{j}~{}\text{for
some}~{}j\leq i\\}.$

Note that $V(T_{i})=LV(T_{i})\cup SV(T_{i})$.

Operation $\mathcal{O}_{1}$. If $x\in LV(T_{i})$, then $\mathcal{O}_{1}$ adds
a double star $DS_{r,s}$ with a center $u$ and joins $u$ to $x$ to produce
$T_{i+1}$, where $s\geq 2$ and $u$ has $r$ leaves.

Operation $\mathcal{O}_{2}$. If $x\in SV(T_{i})$, then $\mathcal{O}_{2}$ adds
a star $K_{1,t}$ with the center $u$ and joins $u$ to $x$ to produce
$T_{i+1}$.

The following is obtained by the induction.

######  Observation 2.1.

With the previous notation, the following holds.

  1. (i)

$LV(T_{i})$ is a unique minimum RDS of $T_{i}$.

  2. (ii)

The subgraph induced by $SV(T_{i})$ is a forest and each component has at
least two vertices.

######  Lemma 2.2.

If $\gamma_{r}(T_{i})=\gamma_{rI}(T_{i})$ and $T_{i+1}$ is obtained from
$T_{i}$ by operation $\mathcal{O}_{1}$, then
$\gamma_{r}(T_{i+1})=\gamma_{rI}(T_{i+1})$.

###### Proof.

It follows from Observation 2.1 that
$\gamma_{r}(T_{i+1})=\gamma_{r}(T_{i})+r+s$. Since every
$\gamma_{rI}(T_{i})$-function can be extended to a RIDF of $T_{i+1}$, we have
$\gamma_{rI}(T_{i+1})\leq\gamma_{rI}(T_{i})+r+s$.

We verify $\gamma_{rI}(T_{i+1})=\gamma_{rI}(T_{i})+r+s$. Let $g$ be a
$\gamma_{rI}(T_{i+1})$-function. If $g(x)=0$, then $g(y)=1$ for each $y\in
N_{T_{i+1}}(x)$. This implies that
$\gamma_{rI}(T_{i+1})\geq\gamma_{rI}(T_{i})+r+s+2$, a contradiction. Thus, we
have $g(x)=1$. It is easy to see that $g|_{V(T_{i})}$ is a RIDF. So, we have
$\gamma_{rI}(T_{i})\leq\gamma_{rI}(T_{i+1})-r-s$.

Thus, it follows from $\gamma_{r}(T_{i})=\gamma_{rI}(T_{i})$ that
$\gamma_{r}(T_{i+1})=\gamma_{rI}(T_{i+1})$. ∎

######  Lemma 2.3.

If $\gamma_{r}(T_{i})=\gamma_{rI}(T_{i})$ and $T_{i+1}$ is obtained from
$T_{i}$ by operation $\mathcal{O}_{2}$, then
$\gamma_{r}(T_{i+1})=\gamma_{rI}(T_{i+1})$.

###### Proof.

It follows from Observation 2.1 that
$\gamma_{r}(T_{i+1})=\gamma_{r}(T_{i})+t$. Since every
$\gamma_{rI}(T_{i})$-function can be extended to a RIDF of $T_{i+1}$, we have
$\gamma_{rI}(T_{i+1})\leq\gamma_{rI}(T_{i})+t$.

We verify $\gamma_{rI}(T_{i+1})=\gamma_{rI}(T_{i})+t$. Let $g$ be a
$\gamma_{rI}(T_{i+1})$-function. If $g(x)=1$, then $g(u)=1$. This implies that
$\gamma_{rI}(T_{i+1})\geq\gamma_{rI}(T_{i})+t+1$, a contradiction. Thus, we
have $g(x)=0$. It is easy to see that $g|_{V(T_{i})}$ is a RIDF. So, we have
$\gamma_{rI}(T_{i})\leq\gamma_{rI}(T_{i+1})-t$.

Thus, it follows from $\gamma_{r}(T_{i})=\gamma_{rI}(T_{i})$ that
$\gamma_{r}(T_{i+1})=\gamma_{rI}(T_{i+1})$. ∎

Now we are ready to prove our main theorem.

######  Theorem 2.4.

A tree $T$ of order $n\geq 3$ is a $(\gamma_{r},\gamma_{rI})$-tree if and only
if $T\in\mathcal{H}\cup\\{K_{1,t}\mid t\geq 2\\}$.

###### Proof.

First, we prove that if $T\in\mathcal{H}\cup\\{K_{1,t}\mid t\geq 2\\}$, then
$\gamma_{rI}(T)=\gamma_{r}(T)$. Clearly,
$\gamma_{rI}(K_{1,t})=\gamma_{r}(K_{1,t})$. Assume that $T\in\mathcal{H}$.
Then there exist a sequence $T_{1},T_{2},\dotsc,T_{m}=T$ $(m\geq 1)$ such that
$T_{1}$ is a double star $DS_{r,s}$, and if $m\geq 2$, $T_{i+1}$ can be
obtained recursively from $T_{i}$ by an operation $\mathcal{O}_{1}$ or
$\mathcal{O}_{2}$ for $1\leq i\leq m-1$. We use induction on $m$. Clearly,
$\gamma_{rI}(T_{1})=\gamma_{r}(T_{1})$. Suppose that the statement is true for
any tree constructed by $m-1$ operations. Let $T^{\prime}=T_{m-1}$. By the
induction hypothesis, $\gamma_{rI}(T^{\prime})=\gamma_{r}(T^{\prime})$. It
follows from Lemma 2.2 or 2.3 that $\gamma_{rI}(T)=\gamma_{r}(T)$.

Next, we prove that if $\gamma_{rI}(T)=\gamma_{r}(T)$, then
$T\in\mathcal{H}\cup\\{K_{1,t}\mid t\geq 2\\}$. We proceed by induction on the
order $n$ of $T$ satisfying $\gamma_{rI}(T)=\gamma_{r}(T)$. Suppose that
$diam(T)=2$. Then $T$ is a star and clearly $\gamma_{rI}(T)=\gamma_{r}(T)$
Thus, $T\in\\{K_{1,t}\mid t\geq 2\\}$. Suppose that $diam(T)=3$. Then $T\cong
DS_{r,s}$ for $r,s\geq 2$. In this case, $T$ can be obtained from $K_{1,r}$ by
operation $\mathcal{O}_{2}$. Hence, we may assume that $diam(T)\geq 4$.

Among all of diametrical paths in $T$, we choose $x_{0}x_{1}\dotsc x_{d}$ so
that it maximizes the degree of $x_{d-1}$. Root $T$ at $x_{0}$. Let
$g=(V_{0}^{g},V_{1}^{g},V_{2}^{g})$ be a $\gamma_{rI}(T)$-function.

Claim 1. $V_{2}^{g}=\emptyset$ and $V_{1}^{g}$ is a RDS of $T$.

Since $V_{1}^{g}\cup V_{2}^{g}$ is a RDS of $T$, we have

$\gamma_{r}(T)\leq|V_{1}^{g}\cup
V_{2}^{g}|=|V_{1}^{g}|+|V_{2}^{g}|\leq|V_{1}^{g}|+2|V_{2}^{g}|=\gamma_{rI}(T).$

Since $\gamma_{rI}(T)=\gamma_{r}(T)$, we must have the equality throughout the
above inequality chain. Thus, $V_{2}^{g}=\emptyset$ and $V_{1}^{g}$ is a RDS
of $T$.

Claim 2. $deg_{T}(x_{d-1})\geq 3$.

Suppose to the contrary that $deg_{T}(x_{d-1})=2$. Suppose that
$deg_{T}(x_{d-2})=2$. In this case, $g(x_{d-1})=1$ for otherwise $g(x_{d-2})$
must be assigned the weight $1$ but this contradicts the fact that $V_{1}^{g}$
is a RDS of $T$. By the same argument, we have $g(x_{d-2})=1$. If
$g(x_{d-3})=1$, then $V_{1}^{g}\setminus\\{x_{d-2},x_{d-1}\\}$ is a RDS with
the cardinality less than $\gamma_{r}(T)$, a contradiction. Thus,
$g(x_{d-3})=0$ and $\sum_{x\in N_{T}(x_{d-3})}g(x)\geq 2$. This implies that
$V_{1}^{g}\setminus\\{x_{d-2}\\}$ is a RDS of $T$. This is a contradiction.

Suppose that $deg_{T}(x_{d-2})\geq 3$. Then each $x\in
N_{T}(x_{d-2})\setminus\\{x_{d-3}\\}$ is either a leaf or a weak stem by
$deg_{T}(x_{d-1})=2$ and the hypothesis about $deg_{T}(x_{d-1})$. If
$g(x_{d-2})=1$, then every vertex of $T_{x_{d-2}}$ has the weight $1$. Let $M$
be the subset of $V_{1}^{g}$ obtained by removing $x_{d-2}$ and weak stems in
$T_{x_{d-2}}$. Now we have $g(x_{d-3})=0$ for otherwise $M$ is a RDS of $T$, a
contradiction. Since $\sum_{x\in N_{T}(x_{d-3})}g(x)\geq 2$, $M$ is a RDS of
$T$, a contradiction. This completes the proof of claim.

Claim 3. $deg_{T}(x_{d-2})\geq 3$.

Suppose to the contrary that $deg_{T}(x_{d-2})=2$. Suppose that
$g(x_{d-2})=0$. Then $g(x_{d-3})=g(x_{d-1})=1$, since $\sum_{x\in
N_{T}(x_{d-2})}g(x)\geq 2$. This is a contradiction by Claim 1.

Suppose that $g(x_{d-2})=1$. Then $g(x_{d-1})=1$. Let $N$ be the subset of
$V_{1}^{g}$ obtained by removing $x_{d-2}$ and $x_{d-1}$. If $g(x_{d-3})=1$,
then $N$ is a RDS of $T$, a contradiction. Thus, we have $g(x_{d-3})=0$. But,
since $\sum_{x\in N_{T}(x_{d-3})}g(x)\geq 2$, $N\cup\\{x_{d-2}\\}$ is a RDS of
$T$, a contradiction. This completes the proof of claim.

We divide our consideration into two cases.

Case 1. $x_{d-3}\in V_{1}^{g}$. Then $x_{d-2},x_{d-1}\in V_{0}^{g}$ for
otherwise the subset of $V_{1}^{g}$ obtained by removing $x_{d-2}$ and stems
in $T_{x_{d-2}}$ is a RDS of $T$, a contradiction.

Subcase 1.1. $x_{d-2}$ has stems except for $x_{d-1}$. Consider
$T_{x_{d-1}}\cong K_{1,t}$ and let $T^{\prime}=T-T_{x_{d-1}}$. Since every
$\gamma_{r}(T^{\prime})$-set (resp., $\gamma_{rI}(T^{\prime})$-function) can
be extended to a RDS (resp., RIDF) of $T$,
$\gamma_{r}(T)\leq\gamma_{r}(T^{\prime})+t$ and
$\gamma_{rI}(T)\leq\gamma_{rI}(T^{\prime})+t$. Since $V_{1}^{g}\setminus
C(x_{d-1})$ (resp., $g|_{V(T^{\prime})}$) is a RDS (resp., RIDF) of
$T^{\prime}$, $\gamma_{r}(T^{\prime})\leq\gamma_{r}(T)-t$ and
$\gamma_{rI}(T^{\prime})\leq\gamma_{rI}(T)-t$. Thus, it follows from
$\gamma_{rI}(T)=\gamma_{r}(T)$ that
$\gamma_{r}(T^{\prime})=\gamma_{rI}(T^{\prime})$. Applying the inductive
hypothesis to $T^{\prime}$, $T^{\prime}\in\mathcal{H}$. By operation
$\mathcal{O}_{2}$, we have $T\in\mathcal{H}$.

Subcase 1.2. $x_{d-2}$ has no stem except for $x_{d-1}$. Consider
$T_{x_{d-2}}\cong DS_{r,s}$ and let $T^{\prime}=T-T_{x_{d-2}}$. Since every
$\gamma_{r}(T^{\prime})$-set (resp., $\gamma_{rI}(T^{\prime})$-function) can
be extended to a RDS (resp., RIDF) of $T$, we have
$\gamma_{r}(T)\leq\gamma_{r}(T^{\prime})+r+s$ and
$\gamma_{rI}(T)\leq\gamma_{rI}(T^{\prime})+r+s$. Since $V_{1}^{g}\setminus
D(x_{d-2})$ (resp., $g|_{V(T^{\prime})}$) is a RDS (resp., RIDF) of
$T^{\prime}$, $\gamma_{r}(T^{\prime})\leq\gamma_{r}(T)-r-s$ and
$\gamma_{rI}(T^{\prime})\leq\gamma_{rI}(T)-r-s$. Thus, it follows from
$\gamma_{rI}(T)=\gamma_{r}(T)$ that
$\gamma_{r}(T^{\prime})=\gamma_{rI}(T^{\prime})$. Applying the inductive
hypothesis to $T^{\prime}$, $T^{\prime}\in\mathcal{H}$. By operation
$\mathcal{O}_{1}$, we have $T\in\mathcal{H}$.

Case 2. $x_{d-3}\in V_{0}^{g}$. Then $x_{d-2},x_{d-1}\in V_{0}^{g}$ for
otherwise every vertex in $T_{x_{d-2}}$ belongs to $V_{1}^{g}$. Since
$x_{d-3}$ is adjacent to at least one vertex not in $T_{x_{d-2}}$, the subset
of $V_{1}^{g}$ obtained by removing $x_{d-2}$ and stems in $T_{x_{d-2}}$ is a
RDS of $T$, a contradiction.

Consider $T_{x_{d-1}}\cong K_{1,t}$ and let $T^{\prime}=T-T_{x_{d-1}}$. Since
every $\gamma_{r}(T^{\prime})$-set (resp., $\gamma_{rI}(T^{\prime})$-function)
can be extended to a RDS (resp., RIDF) of $T$,
$\gamma_{r}(T)\leq\gamma_{r}(T^{\prime})+t$ and
$\gamma_{rI}(T)\leq\gamma_{rI}(T^{\prime})+t$. Since $V_{1}^{g}\setminus
C(x_{d-1})$ (resp., $g|_{V(T^{\prime})}$) is a RDS (resp., RIDF) of
$T^{\prime}$, $\gamma_{r}(T^{\prime})\leq\gamma_{r}(T)-t$ and
$\gamma_{rI}(T^{\prime})\leq\gamma_{rI}(T)-t$. Thus, it follows from
$\gamma_{rI}(T)=\gamma_{r}(T)$ that
$\gamma_{r}(T^{\prime})=\gamma_{rI}(T^{\prime})$. Applying the inductive
hypothesis to $T^{\prime}$, $T^{\prime}\in\mathcal{H}$. By operation
$\mathcal{O}_{2}$, we have $T\in\mathcal{H}$. ∎

##  3\. Restrained Italian trees

In this section, we characterize the trees for which
$\gamma_{rI}(T)=2\gamma_{r}(T)$. First, we introduce a family $\mathcal{F}$ of
trees that can be obtained from a sequence $T_{1},T_{2},\dotsc,T_{m}$ $(m\geq
1)$ of trees such that $T_{1}$ is a path $P_{4}$, and if $m\geq 2$, $T_{i+1}$
can be obtained recursively from $T_{i}$ by one of the following operations
for $1\leq i\leq m-1$.

Define

$LV(T_{i})=\\{v\in V(T_{i})\mid v~{}\text{is a leaf of}~{}T_{j}~{}\text{for
some}~{}j\leq i\\}.$

Operation $\mathcal{O}_{1}$. If $x\in LV(T_{i})$, then $\mathcal{O}_{1}$ adds
a path $P_{3}$ with a leaf $u$ and joins $u$ to $x$ to produce $T_{i+1}$.

Operation $\mathcal{O}_{2}$. If $x\in LV(T_{i})$, then $\mathcal{O}_{2}$ adds
a healthy spider $S_{t,t}$ with the center $u$ and joins $u$ to $x$ to produce
$T_{i+1}$.

Since the family $\mathcal{F}$ is a subclass of $\mathcal{T}$ given in [2,
Lemma 3], we can get the following result.

######  Lemma 3.1.

With the previous notation, the following properties hold.

  1. (i)

$LV(T_{i})$ is a packing.

  2. (ii)

Every $v\in V(T_{i})\setminus LV(T_{i})$ is adjacent to at least one vertex in
$V(T_{i})\setminus LV(T_{i})$ and to exactly one vertex in $LV(T_{i})$.

  3. (iii)

$LV(T_{i})$ is a $\gamma(T_{i})$-set.

  4. (iv)

$LV(T_{i})$ is the unique $\gamma_{r}(T_{i})$-set.

  5. (v)

$LV(T_{i})$ is the unique $\rho(T_{i})$-set.

######  Lemma 3.2.

With the previous notation, $(V(T_{i})\setminus
LV(T_{i}),\emptyset,LV(T_{i}))$ is a $\gamma_{rI}(T_{i})$-function.

###### Proof.

We show that every RIDF has weight at least $2|LV(T_{i})|$. Let $f$ be a RIDF
of $T_{i}$ and $P(T_{i}):=\\{N_{T_{i}}[v]\mid v\in LV(T_{i})\\}$. It follows
from Lemma 3.1 that $P(T_{i})$ is a partition of $V(T_{i})$.

We claim that $f(U)\geq 2$ for each $U\in P(T_{i})$. For a leaf $v\in
V(T_{i})$, clearly $f(N_{T_{i}}[v])=2$. Suppose to the contrary that
$f(N_{T_{i}}[u])=1$ for some $u\in LV(T_{i-1})$. For $w\in N_{T_{i}}(u)$, if
$f(w)=1$, then $u$ is not dominated, a contradiction. Assume that $f(u)=1$.
From the construction of $T_{i}$, there exist at least one vertex $z\in
N_{T_{i}}(u)$ such that $deg_{T_{i}}(z)=2$. To dominate $z$, there must be one
vertex with weight at least one. This is not restrained, a contradiction.
Thus, $\gamma_{rI}(T_{i})\geq 2|LV(T_{i})|=2\gamma_{r}(T_{i})$ and clearly
$(V(T_{i})\setminus LV(T_{i}),\emptyset,LV(T_{i}))$ is a
$\gamma_{rI}(T_{i})$-function. ∎

Now we are ready to prove our main theorem.

######  Theorem 3.3.

A tree $T$ of order $n\geq 4$ is a restrained Italian tree if and only if
$T\in\mathcal{F}$.

###### Proof.

The sufficiency follows from Lemmas 3.1 and 3.2. To prove the necessity, we
proceed by induction on the order $n$ of $T$ satisfying
$\gamma_{rI}(T)=2\gamma_{r}(T)$. It suffices to consider trees with diameter
at least three. Suppose that $diam(T)=3$. Then $T\cong DS_{r,s}$. Since
$\gamma_{r}(T)=\gamma_{rI}(T)=r+s$ for $r,s\geq 2$, we have $DS_{1,1}\cong
P_{4}\in\mathcal{F}$. Hence, we assume that $diam(T)\geq 4$ and $n\geq 5$.

Among all of diametrical paths in $T$, we choose $x_{0}x_{1}\dotsc x_{d}$ so
that it maximizes the degree of $x_{d-1}$. Root $T$ at $x_{0}$. Let $D$ be a
$\gamma_{r}(T)$-set and $g$ be a $\gamma_{rI}(T)$-function defined by $g(v)=2$
for $v\in D$ and $g(u)=0$ for $u\in V(T)\setminus D$.

Claim 1. $x_{d-1}\not\in D$.

Suppose to the contrary that $x_{d-1}\in D$. Since $x_{d}\in D$ and
$\gamma_{rI}(T)=2\gamma_{r}(T)$, we can define a function
$f:V(T)\rightarrow\\{0,1,2\\}$ by $f(x_{d})=1$ and $f(x)=g(x)$ otherwise. Then
$f$ is a RIDF of $T$ with weight less than $\omega(g)$, a contradiction.

Claim 2. $deg_{T}(x_{d-1})=2$.

Suppose to the contrary that $deg_{T}(x_{d-1})\geq 3$. Then there exists at
least one leaf $u\in N_{T}(x_{d-1})\setminus\\{x_{d}\\}$. Since $u,x_{d}\in D$
and $\gamma_{rI}(T)=2\gamma_{r}(T)$, we can define
$f:V(T)\rightarrow\\{0,1,2\\}$ by $f(u)=f(x_{d})=1$ and $f(x)=g(x)$ otherwise.
Then $f$ is a RIDF of $T$ with weight less than $\omega(g)$, a contradiction.

We divide our consideration into two cases.

Case 1. $deg_{T}(x_{d-2})\geq 3$. By Claim 2, each $x\in
N_{T}(x_{d-2})\setminus\\{x_{d-3}\\}$ is either a leaf or a weak stem. Now we
show that $x_{d-2}$ has no leaf. Suppose to the contrary that there exists a
leaf $u\in N_{T}(x_{d-2})$. Then $u\in D$. Note that $x_{d-1},x_{d-2}\not\in
D$. If $x_{d-3}\in D$, then define $f:V(T)\rightarrow\\{0,1,2\\}$ by $f(u)=1$
and $f(x)=g(x)$ otherwise. Clearly $f$ is a RIDF of $T$ with weight less than
$\omega(g)$, a contradiction. Suppose that $x_{d-3}\not\in D$. Define
$h:\rightarrow\\{0,1,2\\}$ by $h(u)=h(x_{d-1})=h(x_{d})=1$ and $h(x)=g(x)$
otherwise. Clearly $h$ is a RIDF of $T$ with weight less than $\omega(g)$, a
contradiction. Thus, $x_{d-2}$ has no leaf and so $T_{x_{d-2}}$ is a healthy
spider.

Since $x_{d-2}$ and its children do not belong to the $\gamma_{r}(T)$-set $D$,
$x_{d-3}$ belongs to $D$. Consider the tree $T^{\prime}:=T-T_{x_{d-2}}$. It is
easy to see that $V(T^{\prime})\cap D$ is a $\gamma_{r}(T^{\prime})$-set and
$\gamma_{rI}(T^{\prime})=2\gamma_{r}(T^{\prime})$. Applying the inductive
hypothesis to $T^{\prime}$, we have $T^{\prime}\in\mathcal{F}$. Since
$x_{d-3}$ is a leaf in $T^{\prime}$, the tree $T$ can be obtained from the
tree $T^{\prime}$ by applying operation $\mathcal{O}_{2}$. Thus,
$T\in\mathcal{F}$.

Case 2. $deg_{T}(x_{d-2})=2$. Then $x_{d-3}\in D$. Consider the tree
$T^{\prime}:=T-T_{x_{d-2}}$ It is easy to see that $V(T^{\prime})\cap D$ is a
$\gamma_{r}(T^{\prime})$-set and
$\gamma_{rI}(T^{\prime})=2\gamma_{r}(T^{\prime})$. Applying the inductive
hypothesis to $T^{\prime}$, we have $T^{\prime}\in\mathcal{F}$. Since
$x_{d-3}$ is a leaf in $T^{\prime}$, the tree $T$ can be obtained from the
tree $T^{\prime}$ by applying operation $\mathcal{O}_{1}$. Thus,
$T\in\mathcal{F}$. ∎

##  4\. Open problems

In this section, we discuss few open problems related to our results. For any
graph theoretical parameters $\sigma$ and $\delta$, we define a tree $T$ to be
$(\sigma,\delta)$-tree if $\sigma(T)=\delta(T)$. In general, it holds
$\gamma_{I}(G)\leq\gamma_{rI}(G)$ for any graph $G$. We suggest the following
problem.

######  Problem 4.1.

Characterize $(\gamma_{I},\gamma_{rI})$-trees.

In [5], D. Ma et al. gave the concept of total restrained domination.
Combining the properties of Italian dominating function and total restrained
dominating set, we give the concept of total restrained Italian dominating
function, namely a RIDF is a total restrained Italian dominating function on
$G$ if the subgraph induced by $\\{v\in V\mid f(v)\geq 1\\}$ has no isolated
vertices. We denote the total restrained Italian domination number by
$\gamma_{rI}^{t}(G)$. The total Italian domination number and total restrained
domination number are denoted by $\gamma_{I}^{t}(G)$ and $\gamma_{r}^{t}(G)$,
respectively (see [4, 5] for definitions). We suggest the following problems.

######  Problem 4.2.

Characterize $(\gamma_{I}^{t},\gamma_{rI}^{t})$-trees.

######  Problem 4.3.

Characterize $(\gamma_{rI},\gamma_{rI}^{t})$-trees.

######  Problem 4.4.

Characterize $(\gamma_{r}^{t},\gamma_{rI}^{t})$-trees.

## References

  * [1] J.A. Bondy, U.S.R. Murty, Graph theory, Graduate Texts in Mathematics 244, Springer, 2007.
  * [2] P. Dankelmann, J.H. Hattingh, M.A. Henning, H.C. Swart, Trees with equal domination and restrained domination numbers, J. Global Optim. 34 (2006) 597–607.
  * [3] G.S. Domke, J.H. Hattingh, S.T. Hedetniemi, R.C. Laskar, L.R. Markus, Restrained domination in graphs, Discrete Math. 203 (1999) 61–69.
  * [4] S.C. Garc$\acute{i}$a, A.C. Mart$\acute{i}$nez, F.A. Mira, I.G. Yero, Total Roman $\\{2\\}$-domination in graphs, Quaest. Math. to appear.
  * [5] D. Ma, X. Chen, L. Sun, On the total restrained domination in graphs, Czechoslovak Math. J. 55 (2005) 165–173.
  * [6] B. Samadi, M. Alishahi, I. Masoumi, D.A. Mojdeh, Restrained Italian domination in graphs, arXiv:2009.12209.
  * [7] D.B. West, Introduction to graph theory, Prentice Hall, Inc., Upper Saddle River, NJ, 2001.