# Video Instance Segmentation in an Open-World

Omkar Thawakar1 Sanath Narayan2 Hisham Cholakkal1 Rao Muhammad Anwer1,3
Salman Khan1 Jorma Laaksonen3 Mubarak Shah4 Fahad Shahbaz Khan1,5
1Mohamed bin Zayed University of AI, UAE 2Technology Innovation Institute, UAE
3Aalto University, Finland 4University of Central Florida, USA 5Linköping
University, Sweden

###### Abstract

Existing video instance segmentation (VIS) approaches generally follow a
closed-world assumption, where only seen category instances are identified and
spatio-temporally segmented at inference. Open-world formulation relaxes the
close-world static-learning assumption as follows: (a) first, it distinguishes
a set of known categories as well as labels an unknown object as ‘unknown’ and
then (b) it incrementally learns the class of an unknown as and when the
corresponding semantic labels become available. We propose the first open-
world VIS approach, named OW-VISFormer, that introduces a novel feature
enrichment mechanism and a spatio-temporal objectness (STO) module. The
feature enrichment mechanism based on a light-weight auxiliary network aims at
accurate pixel-level (unknown) object delineation from the background as well
as distinguishing category-specific known semantic classes. The STO module
strives to generate instance-level pseudo-labels by enhancing the foreground
activations through a contrastive loss. Moreover, we also introduce an
extensive experimental protocol to measure the characteristics of OW-VIS. Our
OW-VISFormer performs favorably against a solid baseline in OW-VIS setting.
Further, we evaluate our contributions in the standard fully-supervised VIS
setting by integrating them into the recent SeqFormer, achieving an absolute
gain of 1.6% AP on Youtube-VIS 2019 val. set. Lastly, we show the
generalizability of our contributions for the open-world detection (OWOD)
setting, outperforming the best existing OWOD method in the literature. Code,
models along with OW-VIS splits are available at
https://github.com/OmkarThawakar/OWVISFormer.

##  1 Introduction

Video instance segmentation (VIS) strives to simultaneously classify, segment
and track all object instances from a set of semantic classes in a given
video. The problem is challenging since a diverse set of objects are desired
to be accurately tracked and segmented despite real-world issues such as, fast
motion, large intra-class variation and background clutter. Most existing VIS
approaches [26, 25, 29, 14] typically follow a close-world assumption, _i.e_.,
all object categories to be detected are provided during the training and only
seen object classes are spatio-temporally segmented at inference. _E.g_.,
existing VIS methods evaluated on the popular Youtube-VIS benchmark [28, 27]
assume that annotated (known) instances of all 40 semantic categories to be
segmented and tracked are available during training. Here, such a training
scheme treats unannotated (unknown) objects as background. Therefore, the
closed-world assumption poses issues to existing VIS methods when recognizing
novel (unknown) object class instances.

The open-world problem formulation [13, 10] relaxes the closed-world
assumption by enabling the VIS model at each training episode to identify
unknown object category instances as belonging to the ‘unknown’ class while
simultaneously learning to spatio-temporally segment a given set of ‘known’
objects. Afterwards, these identified unknowns can be passed to an oracle,
which annotates a set of object categories of interest. Then, the VIS model
takes into account these new knowns to incrementally update its knowledge
without requiring to be retrained from scratch using the previous known object
categories. However, such an open-world formulation poses additional issues
over the standard VIS problem challenges by requiring the model to also (i)
distinguish unknown objects and (ii) recognize them later with the arrival of
progressive training data, in a unified manner. Although the open-world
setting has been explored recently for detection [13, 10] and image
segmentation [16, 23], to the best of our knowledge, we are the first to
investigate the problem and introduce a novel approach for open-world video
instance segmentation (OW-VIS).

Figure 1: Video instance segmentation in an open-world (OW-VIS) setting
illustrating the first step of identifying ‘known’ and ‘unknown’ objects. For
each example video frame, we show the corresponding objectness map obtained
from backbone features in the case of the baseline (col 2 and 7) and the
spatio-temporal objectness (STO) module output map for our OW-VISFormer (col 3
and 8). Moreover, we show the respective output segmentation mask for each
frame in case of the baseline (col 4 and 9) and OW-VISFormer (col 5 and 10).
In these example videos, the unknown (in red dashed line) objects are three
elephants (on the left), and the duck (on the right). The known objects in
these videos are person (on the left) and dog (on the right). Compared to the
baseline, OW-VISFormer accurately segments all the ‘unknown’ and ‘known’
object instances. Best viewed zoomed in. Additional results are presented in
the supplementary.

When designing an OW-VIS framework, one plausible way is to extend a fully-
supervised VIS approach by introducing a pseudo-labeling scheme to identify
potential unknown objects. These potential pseudo-unknowns along with the
ground-truth known instances can then be utilized to learn a foreground-
background class-agnostic separation as well as performing class-specific
known _vs_. unknown instance classification. Existing fully-supervised VIS
approaches typically employ an ImageNet [22] pre-trained classification
backbone for multi-scale feature extraction to be used in the encoder. The
same features can also be utilized in a bottom-up pseudo-labeling scheme in
OW-VIS. However, such a pre-trained classification-based framework is likely
to struggle in the OW-VIS paradigm (see Fig. 1), where the aim is to
accurately distinguish a class-agnostic unknown object from the background as
well as class-specific known categories at the pixel-level. To achieve such an
accurate pixel-level (unknown) object delineation from the background, we
argue that dedicated shallow features are especially desired to complement the
high-level semantic pre-trained features. Moreover, since the selection of
pseudo-unknowns relies on the activations in the selected feature map, it is
further desired to enhance their strengths in the foreground regions (known
and unknown) for learning better objectness priors.

###  1.1 Contributions

We propose an OW-VIS approach, named OW-VISFormer, that introduces (i) a
feature enrichment mechanism, which aims to better differentiate class-
agnostic foreground _vs_. background as well as aid in class-specific known
_vs_. unknown instance classification and (ii) a spatio-temporal objectness
(STO) module that strives to identify candidate pseudo-unknowns. The feature
enrichment mechanism is based on a light-weight auxiliary network that is
trained from scratch and generates dedicated shallow features to complement
the high-level semantic standard pre-trained features. The resulting extended
features are enriched by the encoder and then use in the STO module. Our STO
module employs a contrastive loss that distinguishes candidate pseudo-unknowns
from the background by enhancing the foreground activations. As a result,
improved video instance mask predictions are obtained for both the known and
unknown classes (see Fig. 1). Furthermore, we introduce carefully curated
open-world splits of Youtube-VIS dataset for a rigorous evaluation of OW-VIS
problem.

Our extensive quantitative and qualitative evaluations demonstrate the
effectiveness of the proposed OW-VISFormer leading to consistent improvement
in performance, compared to the baseline. In addition, we also validate our
proposed contributions in the standard fully-supervised VIS problem setting by
introducing them into the recent SeqFormer [26], achieving an absolute gain of
1.6% in overall AP on the Youtube-VIS 2019 val. set. Lastly, we demonstrate
the generalizability of our two contributions for the open-world detection
(OWOD) problem setting by integrating them into the recent OW-DETR [10]. On
the challenging MS COCO OWOD split, our approach outperforms the recent OW-
DETR on all the tasks for both the ‘known’ and ‘unknown’ categories.

##  2 Open-world Video Instance Segmentation

###  2.1 Problem Formulation

Let $\mathcal{D}^{t}=\\{\mathcal{V}^{t},\mathcal{Y}^{t}\\}$ be a progressive
dataset at time $t$ containing $N_{t}$ videos
$\mathcal{V}^{t}=\\{V_{1},\cdots,V_{N_{t}}\\}$ with corresponding labels
$\mathcal{Y}^{t}=\\{\bm{Y}_{1},\cdots,\bm{Y}_{N_{t}}\\}$. Here,
$V_{i}\in\mathcal{R}^{L_{i}\times 3\times H\times W}$ denotes a video of
length $L_{i}$ frames with spatial resolution $H\times W$, while
$\bm{Y}_{i}=\\{\bm{y}_{1},\cdots,\bm{y}_{K}\\}$ denotes the ground-truth mask
annotations of a set of $K$ object instances present in the video. Here,
$\bm{y}_{j}\in\mathcal{R}^{L_{i}\times h\times w}$ denotes the set of masks
predicted for an object instance $j$ in $L_{i}$ frames of a video $V_{i}$. Let
$\mathcal{K}^{t}=\\{1,2,\cdots,C\\}$ denote the known object categories at
time $t$ and $\mathcal{U}^{t}=\\{C\\!+\\!1,\cdots\\}$ be a set of unknown
classes that are likely to be encountered at test time.

As discussed earlier, our OW-VIS first learns a model $\mathcal{M}^{t}$ that
can spatio-temporally segment an unseen class instance at time $t$ as
belonging to the unknown class (denoted by label $0$) in addition to
segmenting the instances of previously encountered known classes
$\mathcal{K}^{t}$. Next, a set of these unknown instances $\bm{U}^{t}$
identified by $\mathcal{M}^{t}$ are then taken as input to an oracle, which
labels $n$ novel classes of interest and provides new training examples for
the corresponding $n$ classes. Then, these $n$ classes are considered as known
and added to the previously known $C$ classes, such that
$\mathcal{K}^{t+1}=\mathcal{K}^{t}+\\{C+1,\cdots,C+n\\}$. Then,
$\mathcal{M}^{t}$ is incrementally trained to obtain an updated model
$\mathcal{M}^{t+1}$, which can spatio-temporally segment all object instances
belonging to classes in $\mathcal{K}^{t+1}$ without forgetting the previously
learned classes in $\mathcal{K}^{t}$. This cycle of spatio-temporally
segmenting unknown instances and incremental learning of new knowledge
continues over the model’s life-time.

###  2.2 Baseline OW-VIS Framework

We base our approach on the recent fully-supervised (FS) SeqFormer [26]. It
utilizes a standard pre-trained backbone network for multi-scale feature
extraction followed by a deformable transformer [30] and a segmentation block
for video instance mask prediction. Here, an $M$-frame video clip
$\mathbf{v}\subset V_{i}$ is input to a pre-trained backbone network. Then,
the resulting multi-scale features of each frame are input to an encoder that
outputs feature maps of the same size as the input. The encoder output
features together with $q$ learnable instance query embeddings $\bm{Q}^{I}$
are input to the decoder. Consequently, the decoder outputs $q$ instance
features $\bm{F}^{I}$ that are then used for video mask prediction.

A straightforward way to extend the above FS SeqFormer to OW-VIS (Sec. 2.1) is
to introduce a pseudo-labeling scheme for selecting potential unknown objects
followed by learning to categorize these identified pseudo-unknowns into a
single unknown class. One way to design such a pseudo-labeling scheme is to
utilize object proposals111Proposals are obtained from the instance features
$\bm{Q}^{I}$ and only those remaining after selecting the ground-truth class
instances through Hungarian matching are considered. having high activations
in their corresponding regions of the backbone feature maps as candidates for
unknown class, as in [10]. These pseudo-unknowns can be then used along with
ground-truth known instances to learn a foreground-background class-agnostic
separation as well as perform class-specific known _vs_. unknown instance
classification. We refer it as our baseline OW-VIS framework.

Figure 2: (a) Overall architecture of the proposed OW-VISFormer framework. It
comprises a standard pre-trained backbone, a light-weight Scratch-Net, an
encoder-decoder followed by class-specific, class-agnostic and regression
branches along with a spatio-temporal objectness (STO) module. Different from
the pre-trained backbone, (b) the Scratch-Net is trained from scratch with
random weight initialization during the OW-VIS training. Its light-weight
architecture comprises two 3D convolution and normalization layers. The
resulting feature maps are then integrated with the standard backbone features
to construct extended multi-scale features which are then input to the encoder
that outputs enriched features. Within the decoder, the instance queries
cross-attend to the enriched features. The instance and box features output by
the decoder are input to the class-specific (multi-class), class-agnostic and
regression branches. To effectively learn the class-agnostic and class-
specific branches with the unknown instances in the OW-VIS setting, we
introduce a (c) spatio-temporal objectness (STO) module trained with a
contrastive loss ($\mathcal{L}_{contr}$) for generating instance-level pseudo-
labels. Consequently, the instance features from the decoder along with the
encoder features are used within the segmentation block for the video instance
mask prediction of known and unknown classes in the proposed OW-VISFormer
framework.

##  3 Proposed OW-VIS Framework

Overall Architecture: Fig. 2(a) shows the overall OW-VISFormer framework with
a standard pre-trained backbone. To circumvent using a fully-supervised
ImageNet pre-trained network, we use the popular self-supervised ResNet50 DINO
[7] ImageNet-1K backbone. For accurate pixel-level object delineation from the
background, we introduce a feature enrichment mechanism that utilizes a novel
light-weight ScratchNet Fig. 2(b) which is trained from scratch through random
weight initialization. Our ScratchNet, comprising two 3D convolution and
normalization layers, takes the same input frames as the standard pre-trained
backbone stream. The resulting shallow features from ScratchNet are integrated
with high-level semantic features from the standard pre-trained backbone to
produce extended multi-scale features. These extended features are then input
to the encoder to obtain enriched multi-scale features. The decoder within our
OW-VISFormer framework aggregates the enriched features from all encoder
layers, which are then cross-attended with the instance queries $\bm{Q}^{I}$.
The decoder then outputs the instance and box features ($\bm{F}^{I}$ and
$\bm{F}^{B}$), which are input to the three branches: class-specific, class-
agnostic and regression. As discussed earlier, in the OW-VIS setting both the
class-specific as well as the class-agnostic branches are required to be
learned with the unknown instances. To this end, we propose a spatio-temporal
objectness (STO) module, shown in Fig. 2(c), comprising a 3D convolution layer
for spatio-temporally aggregating the objectness information of video
instances across multiple frames $M$. The STO module is trained with a
contrastive loss ($\mathcal{L}_{contr}$) and enables improved foreground-
background separability resulting in a better unknown instance mask
prediction. Finally, the instance features from the decoder and enriched
multi-scale features from the encoder are utilized within the segmentation
block to produce video instance mask predictions for both the known and
unknown classes. Next, we describe our feature enrichment mechanism.

###  3.1 Feature Enrichment Mechanism

Light-weight ScratchNet: the OW-VIS setting requires accurate pixel-level
(unknown) object delineation from the background as well as distinguishing
category-specific known classes. Therefore, shallow features capturing
distinct edge and boundary information are desired to complement the high-
level semantic features. A straightforward strategy to integrate the shallow
feature information is to re-use the initial layer features from the standard
pre-trained backbone. However, we empirically observe this to achieve inferior
performance compared to the features generated using the proposed ScratchNet.
We conjecture that the low-/mid-level shallow features from the initial layers
of the standard pre-trained backbone are better adapted for the task of image
classification, which prefers translation invariance. In contrast, the
proposed light-weight ScratchNet produces dedicated shallow features that aid
in accurate video instance mask prediction for both unknown as well as known
categories. The light-weight ScratchNet is trained from scratch through random
weights initialization. It comprises two layers of 3D convolution, each
followed by a layer normalization, as shown in Fig. 2(b). Both convolutional
layers perform a non-overlapping convolution operation on their inputs with a
kernel size of $4\times 4\times 4$ and a stride of 4. ScratchNet produces
complementary shallow features, which are then integrated as an additional
feature scale along with multi-scale pre-trained backbone features, resulting
in extended multi-scale features.
Enriched Multi-scale Encoder Features: The extended multi-scale features are
combined with positional encodings and input to a deformable encoder [30]
consisting of six layers of multi-scale deformable attention. The standard
deformable encoder in our OW-VIS baseline (Sec. 2.2) outputs the attended
multi-scale features from its final layer alone, which are tailored for known
category mask prediction. This can likely lead to deterioration in some of the
relevant features for accurate mask prediction of unknown instances in our OW-
VIS setting. To alleviate this issue, we combine features from all encoder
layers in order to learn enriched features for predicting accurate video masks
for both known and unknown instances. To this end, multi-scale features from
all but the final encoder layer are scale-wise fused through a convolution
operation and added with the features of the final encoder layer, resulting in
enriched multi-scale features. Such a feature fusion enables improving unknown
video instance mask prediction while preserving the known category
predictions. Consequently, the encoder outputs enriched multi-scale features
that are better suited for the OW-VIS task. These enriched multi-scale
features are then input to the decoder as well as our spatio-temporal
objectness (STO) module described next.

Figure 3: The proposed task composition in our OW-VIS evaluation setting based
on the Youtube-VIS dataset. Here, for each task in the corresponding split we
show the number of videos and instances (objects). We construct five splits
(A-E) each having two tasks (Task-1 and Task-2). For Task-1, the super-
categories in Split A, B, C, D and E are: {Human, Animals, Aquatic Animals},
{Human, Vehicle and Others}, {Human, Animals}, {Human, Animals (domestic),
Aquatic Animals (amphibious), Vehicle (road) and Others (board)} and {Human,
Animals (wild), Aquatic Animals (underwater), Vehicle (non-road) and Others
(non-board)}. Similarly, for Task-2, the super-classes in Split A, B, C, D and
E are: {Vehicle and Others}, {Animals, Aquatic Animals}, {Aquatic Animals,
Vehicle and Others}, {Human, Animals (wild), Aquatic Animals (underwater),
Vehicle (non-road) and Others (non-board)}, {Human, Animals (domestic),
Aquatic Animals (amphibious), Vehicle (road) and Others (board)}.

###  3.2 Spatio-Temporal Objectness Module

In the OW-VIS setting, both category-specific as well as class-agnostic
branches require learning with the unknown instances. To this end, we
introduce a spatio-temporal objectness (STO) module (Fig. 2 (c)) that
generates instance-level pseudo-labels and is trained using a contrastive
loss. The STO module $G_{STO}(\cdot)$ consists of a 3D convolutional layer
with output channels equal to one. It takes the $d$-dimensional enriched
encoder features $\bm{E}_{k}$ corresponding to the $M$ frames at spatial scale
$k=\nicefrac{{1}}{{16}}$ as input and outputs an objectness map
$O_{map}\in\mathcal{R}^{M\times\nicefrac{{H}}{{16}}\times\nicefrac{{W}}{{16}}}$.
Pseudo-labeling Unknown Instances: Given the $q$ instance features
$\bm{F}^{I}$ output by the decoder, we employ the Hungarian matching loss [15]
that identifies the best matching queries for the $K$ known instances in the
input video. For each of the remaining $q-K$ instance predictions, their
objectness scores $s_{i}$ (where $i\in\\{K+1,\cdots,q\\}$) are computed by
spatio-temporally aggregating the activation strengths of the objectness maps
$O_{map}=G_{STO}(\bm{E}_{k})$ within the corresponding predicted box regions
across $m=[1,\cdots,M]$ frames, given by

$s_{i}=\sum_{m=1}^{M}{\frac{1}{h_{i}^{m}\cdot
w_{i}^{m}}}\sum_{x_{i}^{m}-{0.5w_{i}^{m}}}^{x_{i}^{m}+{0.5w_{i}^{m}}}\sum_{y_{i}^{m}-{0.5h_{i}^{m}}}^{y_{i}^{m}+{0.5h_{i}^{m}}}G_{STO}(\bm{E}_{k}),$
(1)

where $\bm{b}_{i}^{m}=[x_{i}^{m},y_{i}^{m},w_{i}^{m},h_{i}^{m}]$ denotes the
box proposal predicted in the regression branch for the $i^{th}$ instance in
$m^{th}$ frame. Here, ($x_{i}^{m},y_{i}^{m}$), $w_{i}^{m}$ and $h_{i}^{m}$
denote the center, width and height, respectively. The resulting $q-K$ scores
($s_{i}$) are sorted in decreasing order and the top-$p_{u}$ instances are
employed as pseudo-unknown during training. Furthermore, the remaining
$q-(K+p_{u})$ instances are considered as background instances.

Given that the selection of pseudo-unknowns depends on the activations in the
objectness map $O_{map}$, we introduce a contrastive loss that aims to better
separate the class-agnostic foreground regions from the background in the
objectness map. The foreground score $S_{fg}$ is computed by aggregating the
objectness scores $s_{i}$ of the foreground video instances, _i.e_.,
$i\in[1,\cdots,K+p_{u}]$ (both known and pseudo-unknown). Similarly, the
background score $S_{bg}$ is obtained by aggregating the objectness scores
$s_{i}$ of the background video instances, _i.e_., $i\in[K+p_{u}+1,\cdots,q]$.
The constrastive loss is then given by

$\displaystyle\mathcal{L}_{contr}\bigl{(}S_{fg},S_{bg}\bigr{)}=e^{-(S_{fg}-S_{bg})},$
(2) $\displaystyle\text{where}\quad
S_{fg}=\sum_{i=1}^{K+p_{u}}s_{i}\quad\text{and}\quad
S_{bg}=\sum_{i=K+p_{u}+1}^{q}s_{i}.$ (3)

The resulting pseudo-unknown and background instances along with the ground-
truth known instances are employed for training the class-specific and class-
agnostic branches. Furthermore, as in [26], the instance and box features
($\bm{F}^{I}$ and $\bm{F}^{B}$) output by the decoder, along with the enriched
multi-scale features $\bm{E}$ are utilized as input to the segmentation block
for predicting the video masks of both known and unknown instances.

###  3.3 Training and Inference

Training: Our proposed OW-VISFormer framework is trained with the loss
formulation given by

$\mathcal{L}=\mathcal{L}_{c}+\mathcal{L}_{r}+\alpha\mathcal{L}_{f}+\mathcal{L}_{contr},$
(4)

where $\mathcal{L}_{c}$, $\mathcal{L}_{f}$, $\mathcal{L}_{r}$ and
$\mathcal{L}_{contr}$, respectively denote the classification (class-specific
branch), foreground objectness (class-agnostic branch), regression (box and
mask) and constrastive (STO module) loss terms. While $\mathcal{L}_{c}$ and
$\mathcal{L}_{f}$ are computed using the focal loss [18], $\mathcal{L}_{r}$ is
the standard $l_{1}$ loss. Furthermore, a balanced set of exemplars is
utilized to finetune the model after the incremental step in each task for
alleviating catastrophic forgetting, as in [10, 13].

Inference: At test time, the class-specific branch predictions of the $C$
known classes are utilized and top-$k$ instances are selected as known
instances. Furthermore, among the remaining $q-k$ instances, top-$k$ with high
unknown class probability are selected as unknown instances. Finally, the box
and instance features from the decoder for the corresponding known and unknown
instances predicted, along with the enriched multi-scale encoder features are
input to the segmentation block for predicting the video masks.

Table 1: Comparison between the baseline and our OW-VISFormer on the five
splits (A-E) introduced for OW-VIS setting. For the Task-1 evaluation, the
results are reported in terms of overall AP and recall (AR-1) for both ‘Known’
classes and the ‘Unknown’ category. For the Task-2, which involves the
incremental learning step, we report the OW-VIS setting results for
‘Previously Known’, ‘Current Known’ along with ’Both’. Note that the ‘Unknown’
class performance is not reported for Task-2 since all 40 classes are ‘Known’.
Our proposed OW-VISFormer achieves consistent gains over the baseline on all
splits across all tasks for both ’Known’ and ‘Unknown’ classes.

Task-1 Task-2 Known Unknown Previously Known Current Known Both AP AR-1 AP
AR-1 AP AR-1 AP AR-1 AP AR-1 Split A Baseline 35.3 34.8 6.7 9.5 30.6 31.2 30.3
30.7 30.4 31.0 Ours 36.7 37.3 10.0 11.9 32.5 33.9 34.1 34.9 33.3 34.4 Split B
Baseline 31.0 31.4 2.7 5.1 28.9 30.5 31.7 32.6 30.3 31.6 Ours 32.2 33.1 6.5
8.7 30.4 32.2 35.1 35.7 32.7 33.9 Split C  Baseline 33.9 33.3 4 7.2 28.7 32.6
32.1 32.6 30.4 28.7 Ours 36.4 35.2 7.1 9.6 30.6 33.2 35.0 35.1 32.8 34.1 Split
D  Baseline 31.4 34.5 3.3 6.4 29.7 30.9 30.2 32.2 30.0 31.6 Ours 33.6 35.0 6.9
9.7 31.7 32.2 33.5 34.2 32.6 33.2 Split E  Baseline 32.0 35.1 3.5 6.5 29.7
31.2 30.4 31.7 30.1 29.7 Ours 35.1 36.3 5.6 8.9 31.3 32.1 33.9 36.0 32.6 34.0

##  4 Experiments

###  4.1 Experiment 1: OW-VIS Setting

Datasets: We adapt the popular Youtube-VIS [27] dataset to construct open-
world VIS (OW-VIS) data splits. For each split, we group the $40$ categories
into two mutually exclusive sets, thereby constructing tasks with non-
overlapping classes $\\{\mathcal{T}_{1},\mathcal{T}_{2}\\}$ such that, the
$\mathcal{T}_{2}$ categories are not known during task $\mathcal{T}_{1}$.
Then, during the learning of task $\mathcal{T}_{t}$, all the category labels
belonging in $\\{\mathcal{T}_{\alpha}:\alpha\leq t\\}$ are considered as
known. Similarly, labels belonging to $\\{\mathcal{T}_{\alpha}:\alpha>t\\}$
are considered as unknown during evaluation. To construct a test set at time
$t$, we consider 20% of videos in tasks $\\{\mathcal{T}_{\alpha}:\alpha\leq
t\\}$ for known evaluation and 20% of videos in tasks
$\\{\mathcal{T}_{\alpha}:\alpha>t\\}$ for unknown evaluation. By grouping
super-categories of all 40 Youtube-VIS categories in different ways, we
created $5$ such splits as shown in Fig. 3. Additional details are provided in
the supplementary.
Evaluation Metric: We adapt the standard VIS evaluation metrics for evaluating
the OW-VIS setting. For the known classes as well as the unknown class, the
standard overall average precision (AP) and average recall (AR) are used.

Implementation Details: We employ the self-supervised DINO [7] ResNet50
Imagenet-1k pre-trained backbone to extract multi-scale features using the
$conv_{3}$, $conv_{4}$ and $conv_{5}$ stages. The resulting multi-scale
features are mapped to the same feature dimension of $256$ through
convolution, as in [26, 30]. The two convolution layers in ScratchNet employ
kernel size $4\times 4$ and stride $4$ along with the output channels set to
$256$. The encoder and decoder are six layers each with latent dimension being
$256$. The number of instance queries $q$ is set to $300$. Our OW-VISFormer is
implemented in PyTorch-1.8 [21] and learned using the Adam optimizer with
learning rate set to $10^{-4}$. The model is trained on Task-1 data for $18$
epochs. For Task-2, the learned model from Task-1 is incrementally trained via
the memory replay. Specifically, it is trained on current known classes of
Task-2 for $12$ epochs and then finetuned for $2$ epochs by including previous
known class exemplars. Our code, models and data splits will be made public.

Quantitative Comparison: We compare the performance of the baseline (2.2) and
our OW-VISFormer (3) on the OW-VIS splits (see Fig. 3). Tab. 1 shows the
comparison on all five splits and the corresponding two tasks in each split.
We report the performance in terms of AP and recall for both known and
unknown. For the Task-1 in split A, the baseline achieves a known class AP of
35.3% and AR-1 score of 34.8%. Our OW-VISFormer achieves consistent
improvement in performance in terms of both AP and AR-1 by achieving 36.7% and
37.3%, respectively. Notably, OW-VISFormer obtains a considerable gain in
performance in the case of unknown class, in terms of both AP and AR-1, owing
to the proposed feature enrichment mechanism and the STO module. When the
unknown class labels are progressively labeled in Task-2, we observe OW-
VISFormer to better maintain both objectives of (i) spatio-temporally
segmenting the new known categories and (ii) not forgetting the previously
known classes. Moreover, we observe a consistent improvement in performance
from OW-VISFormer over the baseline for other splits.

Figure 4: Qualitative OW-VIS results on example video frames from the the test
sets of different splits. Here, for each example video, we show the
segmentation masks obtained from our OW-VISFormer when trained only on Task-1
categories (row 1 and 2). The instance mask predictions for the same video
frames are shown after incrementally learning with Task-2 categories (row 3
and 4). From left to right: the unknown objects are encircled by red dashed
lines in the input video frames during the Task-1 evaluation. The unknown
objects are accurately segmented first as ‘unknown’ in Task-1 and then later
correctly classified and spatio-temporally segmented into their respective
‘known’ classes during Task-2 evaluation. Best viewed zoomed in. Additional
results are in the supplementary.  Table 2: State-of-the-art comparison on
YouTube-VIS 2019 val set. All results are reported using the same ResNet-50
backbone. Our approach outperforms the recent SeqFormer [26] with an absolute
gain of 1.6% in terms of overall AP. Best results are in bold.

Method | AP | AP50 | AP75 | AR1 | AR10
---|---|---|---|---|---
MaskTrack R-CNN[28] | $30.3$ | $51.1$ | $32.6$ | $31.0$ | $35.5$
SipMask-VIS[5] | $32.5$ | $53.0$ | $33.3$ | $33.5$ | $38.9$
VisTR[25] | $36.2$ | $59.8$ | $36.9$ | $37.2$ | $42.4$
CrossVIS[29] | $36.6$ | $57.3$ | $39.7$ | $36.0$ | $42.0$
IFC[12] | $42.8$ | $65.8$ | $46.8$ | $43.8$ | $51.2$
DeVIS[4] | $44.4$ | $66.7$ | $48.6$ | $42.4$ | $51.6$
SeqFormer[26] | $45.1$ | $66.9$ | $50.5$ | $45.6$ | $54.1$
Our Approach | 46.7 | 69.1 | 51.7 | 46.1 | 54.9

Ablation Study: We also conduct an experiment by progressively integrating our
contributions into the baseline on one of the difficult splits (split B). For
the known classes, the baseline achieves a AP score of 31.0%. The performance
on the known classes is improved by the introduction of our proposed feature
enrichment mechanism (Sec. 3.1) with an AP score of 31.5%. Furthermore, the
integration of our STO module (Sec. 3.2) that generates instance-level pseudo-
labels improves the AP score to 32.2%, leading to an absolute overall gain of
1.2% over the baseline. For the unknown class, the baseline achieves AP and
AR-1 scores of 2.7% and 5.1%. Integrating the proposed feature enrichment
mechanism significantly improves the performance to 4.5% and 6.1%, in terms of
AP and AR-1. The introduction of the STO module leads to a consistent
improvement in both AP and AR-1, achieving absolute final gains of 3.8% and
3.6% over the baseline. We further perform an experiment to validate the
impact of the dedicated shallow features generated from the proposed
ScratchNet. To this end, we use the shallow features $conv_{2}$ from the
standard pre-trained backbone instead in our OW-VISFormer. The reduces the
performance in terms of both known and unknown AP from 32.2% and 6.5% to 31.2%
and 3.8%. This inferior performance likely suggests that the shallow features
from standard pre-trained backbone are more suited for image classification
task. In contrast, the shallow features obtained through our ScratchNet are
trained from scratch on OW-VIS data and therefore dedicated to aid in accurate
video mask prediction for both unknown as well as known classes.

Qualitative Analysis: Fig. 4 shows the qualitative results from our OW-
VISFormer on example test video frames of different OW-VIS splits. The first
two rows shows the segmentation results obtained on the corresponding video
frames in the Task-1 evaluation (‘known’ and ‘unknown’ objects). The last two
rows show the results from Task-2 evaluation which involves incrementally
training with Task-2 categories. In the Task-1, OW-VISFormer is able to first
accurately segment the different ‘known’ and ‘unknown’ instances. Then, in the
Task-2 evaluation, OW-VISFormer successfully identifies the correct categories
for the same unknown instances that were introduced in the Task-2 learning
while still accurately predicting the instance masks for the previously known
categories from Task-1.

Table 3: State-of-the-art comparison for the open-world object detection
(OWOD) problem on MS COCO split of [10]. The comparison is presented in terms
of unknown category recall (U-Recall) and the known class mAP for the Task-1.
For the remaining tasks (2-4) involving incremental learning steps, we report
the mAP scores for ‘Previously Known’, ‘Current Known’ along with ‘Both’.
Furthermore, the U-Recall is only reported for tasks 1-3 since all classes are
known in the final task 4. When using the same backbone, our approach
outperforms the recent OW-DETR on all tasks for both the ‘Known‘ classes and
the ‘Unknown’ class. Best results are in bold.

Task IDs Task-1 Task-2 Task-3 Task-4 U-Recall mAP U-Recall mAP U-Recall mAP
mAP Current Previously Current Both Previously Current Both Previously Current
Both Known Known Known Known Known Known Known ORE-EBUI [13] 1.5 61.4 3.9 56.5
26.1 40.6 3.6 38.7 23.7 33.7 33.6 26.3 31.8 OW-DETR [10] 5.7 71.5 6.2 62.8
27.5 43.8 6.9 45.2 24.9 38.5 38.2 28.1 33.1 Our Approach 8.8 72.1 8.4 63.3
28.1 44.5 9.1 45.4 25.5 38.9 38.7 29.3 34.0

###  4.2 Experiment 2: Fully-supervised VIS Setting

We further evaluate the effectiveness of our proposed contributions (feature
enrichment mechanism and STO module) in the standard full-supervised (FS) VIS
problem setting. Our main intuition is that the proposed contributions can
likely aid in reducing the confusion of an object region (both known and
unknown) being called as a background. Hence, this can serve as an additional
learning step for better object identification. To this end, we integrate the
feature enrichment mechanism (Sec. 3.1) and our STO module (Sec. 3.2) into the
recent SeqFormer [26]. Tab. 2 shows the results in the standard FS setting on
the Youtube-VIS 2019 val. set. Our approach obtained by integrating the
feature enrichment and the STO module within the standard SeqFormer obtains
overall AP of 46.7%, leading to an absolute gain of 1.6% in AP over recent
best [26]. Fig. 5 presents a qualitative comparison of our approach with the
standard SeqFormer. Our approach achieves improved video instance mask
predictions, compared to SeqFormer.

###  4.3 Experiment 3: Open-World Detection Setting

Lastly, we also validate the generalizability of our feature enrichment and
the STO module for open-world object detection (OWOD) in images. The OWOD
problem has recently gained popularity with evaluations being performed on the
challenging MS COCO dataset [19]. To this end, we integrate our feature
enrichment and STO module (replacing 3D convolutions with 2D) into the recent
transformers-based OWOD framework, named OW-DETR [10]. Tab. 3 shows the
results on the challenging MS COCO split introduced in [10]. For a fair
comparison, we employ the same self-supervised ResNet50 backbone as in the
standard OW-DETR [10]. We report the results of ORE-EBUI [13] and our baseline
OW-DETR from [10]. In the case of Task-1, our approach achieves an impressive
performance particularly on the ‘Unknown’ object class by increasing the
U-Recall from 5.7% to 8.8%, while also improving the detection performance on
the ‘Current Known’ classes. Further, the performance on both ‘Unknown‘ and
‘Known’ are consistently improved in the subsequent tasks as well. For the
final task (task-4) involving all 80 classes from MS COCO as ‘Known‘, our
approach improves the performance over the recent OW-DETR in the case of
‘Both’ previously known and unknown classes from 33.1% to 34.0% mAP.

##  5 Relation to Prior Art

Existing video instance segmentation (VIS) methods can be categorized based on
the underlined detection architecture such as, two-stage [28, 17, 3], single-
stage [14, 5, 2, 9, 8] and transformer-based approaches [25, 26, 29, 12, 4,
1]. Most two-stage VIS approaches extend a two-stage detector such as, Mask
R-CNN [11] by integrating a tracking branch. Most single-stage VIS methods
adapt the one-stage pipeline, where the final mask prediction is obtained
through a linear combination of mask bases. Recently, several works explored
transformers-based detection architecture [30, 6] to formulate VIS as a direct
end-to-end sequence prediction. These approaches predominantly solve the
problem following a closed-world assumption, where the annotated instances of
all (known) semantic classes to be spatio-temporally segmented are available
during instance. This assumption posses issues to most existing VIS approaches
when classifying a novel (unknown) object class instance.

Figure 5: Qualitative comparison between the baseline SeqFormer and our
approach on the fully-supervised setting on Youtube-VIS 2019 val. set. Our
approach obtains improved instance mask segmentation, compared to the
baseline. Best viewed zoomed in. Additional results are in the supplementary.

Several recent works have explored the open-world problem setting for
different tasks, such as detection [13, 10], image instance segmentation [16,
23], and object tracking and video segmentation [20, 24]. However, these works
on open-world video object segmentation [20, 24] only look into the task of
unknown identification and does not address the more challenging open-world
problem setting explored in this work, which requires (i) distinguishing a set
of ‘known’ semantic classes as well as identifying an unknown object as
‘unknown’ and then (ii) incrementally learning the class of an ‘unknown’ as
and when the corresponding semantic category labels are available. To the best
of our knowledge, we are the first to investigate and introduce an approach
for the problem of open-world VIS (OW-VIS).

##  6 Conclusions

We proposed an approach, named OW-VISFormer, to address open-world VIS (OW-
VIS). OW-VISFormer introduces a feature enrichment mechanism to produce
enriched features and a spatio-temporal objectness module that generates
instance-level pseudo-labels. Moreover, we propose OW-VIS splits to identify
unknown, segment known and unknown along with progressively segmenting new
semantic classes. OW-VISFormer achieves competitive performance in three
settings: OW-VIS, FS-VIS and OWOD.

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# taylor’s formula and integral inequalities for conformable fractional
derivatives

Douglas R. Anderson  Concordia College, Department of Mathematics, Moorhead,
MN 56562 USA<EMAIL_ADDRESS>
###### Abstract.

We derive Taylor’s Formula for conformable fractional derivatives. This is
then employed to extend some recent and classical integral inequalities to the
conformable fractional calculus, including the inequalities of Steffensen,
$\breve{\text{C}}$eby$\breve{\text{s}}$ev, Hermite-Hadamard, Ostrowski, and
Grüss.

###### Key words and phrases:

Cauchy Function, Variation of Constants, Taylor’s Formula

###### 2010 Mathematics Subject Classification:

26A33

##  1\. Taylor Theorem

We use the conformable $\alpha$-fractional derivative, recently introduced in
[4, 7], which for $\alpha\in(0,1]$ is given by

(1.1) $D_{*}f(t):=\lim_{\varepsilon\rightarrow 0}\frac{f(t+\varepsilon
t^{1-*})-f(t)}{\varepsilon},\quad D_{*}f(0)=\lim_{t\rightarrow
0^{+}}D_{*}f(t).$

Note that if $f$ is differentiable, then

(1.2) $D_{\alpha}f(t)=t^{1-\alpha}f^{\prime}(t),$

where $f^{\prime}(t)=\lim_{\varepsilon\rightarrow
0}[f(t+\varepsilon)-f(t)]/\varepsilon$.

We will consider Taylor’s Theorem in the context of iterated fractional
differential equations. In this setting, the theorem will be proven using the
variation of constants formula, where we use an approach similar to that used
for integer-order derivatives found in [6]. With this in mind, we begin this
note with a general higher-order equation. For $n\in\mbox{\Bbb N}_{0}$ and
continuous functions $p_{i}:[0,\infty)\to\mbox{\Bbb R}$, $1\leq i\leq n$, we
consider the higher-order linear $\alpha$-fractional differential equation

(1.3) $Ly=0,\quad\mbox{ where }\quad
Ly=D^{n}_{\alpha}y+\sum_{i=1}^{n}p_{i}D^{n-i}_{\alpha}y,$

where $D^{n}_{\alpha}y=D^{n-1}_{\alpha}(D_{\alpha}y)$. A function
$y:[0,\infty)\to\mbox{\Bbb R}$ is a solution of equation (1.3) on $[0,\infty)$
provided $y$ is $n$ times $\alpha$-fractional differentiable on $[0,\infty)$
and satisfies $Ly(t)=0$ for all $t\in[0,\infty)$. It follows that
$D^{n}_{\alpha}y$ is a continuous function on $[0,\infty)$.

Now let $f:[0,\infty)\to\mbox{\Bbb R}$ be continuous and consider the non-
homogeneous equation

(1.4) $D^{n}_{\alpha}(t)+\sum_{i=1}^{n}p_{i}(t)D^{n-i}_{\alpha}y(t)=f(t).$

######  Definition 1.1.

We define the Cauchy function $y:[0,\infty)\times[0,\infty)\to\mbox{\Bbb R}$
for the linear fractional equation (1.3) to be, for each fixed
$s\in[0,\infty)$, the solution of the initial value problem

$Ly=0,\quad D^{i}_{\alpha}y(s,s)=0,\quad 0\leq i\leq n-2,\quad
D^{n-1}_{\alpha}y(s,s)=1.$

######  Remark 1.1.

Note that

$y(t,s):=\frac{1}{(n-1)!}\left(\frac{t^{\alpha}-s^{\alpha}}{\alpha}\right)^{n-1}$

is the Cauchy function for $D^{n}_{\alpha}=0$, which can be easily verified
using (1.2).

######  Theorem 1.1 (Variation of Constants).

Let $\alpha\in(0,1]$ and $s,t\in[0,\infty)$. If $f$ is continuous, then the
solution of the initial value problem

$Ly=f(t),\quad D^{i}_{\alpha}y(s)=0,\quad 0\leq i\leq n-1$

is given by

$y(t)=\int_{s}^{t}y(t,\tau)f(\tau)\tau^{\alpha-1}d\tau,$

where $y(t,\tau)$ is the Cauchy function for (1.3).

###### Proof.

With $y$ defined as above and by the properties of the Cauchy function we have

$D^{i}_{\alpha}y(t)=\int_{s}^{t}D^{i}_{\alpha}y(t,\tau)f(\tau)\tau^{\alpha-1}d\tau+D^{i-1}_{\alpha}y(t,t)f(t)=\int_{s}^{t}D^{i}_{\alpha}y(t,\tau)f(\tau)\tau^{\alpha-1}d\tau$

for $0\leq i\leq n-1$, and

$\displaystyle D^{n}_{\alpha}y(t)$ $\displaystyle=$
$\displaystyle\int_{s}^{t}D^{n}_{\alpha}y(t,\tau)f(\tau)\tau^{\alpha-1}d\tau+D^{n-1}_{\alpha}y(t,t)f(t)$
$\displaystyle=$
$\displaystyle\int_{s}^{t}D^{n}_{\alpha}y(t,\tau)f(\tau)\tau^{\alpha-1}d\tau+f(t).$

It follows from these equations that

$D^{i}_{\alpha}y(s)=0,\quad 0\leq i\leq n-1$

and

$Ly(t)=\int_{s}^{t}Ly(t,\tau)f(\tau)\tau^{\alpha-1}d\tau+f(t)=f(t),$

and the proof is complete. ∎

######  Theorem 1.2 (Taylor Formula).

Let $\alpha\in(0,1]$ and $n\in\mbox{\Bbb N}$. Suppose $f$ is $(n+1)$ times
$\alpha$-fractional differentiable on $[0,\infty)$, and $s,t\in[0,\infty)$.
Then we have

$f(t)=\sum_{k=0}^{n}\frac{1}{k!}\left(\frac{t^{\alpha}-s^{\alpha}}{\alpha}\right)^{k}D^{k}_{\alpha}f(s)+\frac{1}{n!}\int_{s}^{t}\left(\frac{t^{\alpha}-\tau^{\alpha}}{\alpha}\right)^{n}D^{n+1}_{\alpha}f(\tau)\tau^{\alpha-1}d\tau$

###### Proof.

Let $g(t):=D^{n+1}_{\alpha}f(t)$. Then $f$ solves the initial value problem

$D^{n+1}_{\alpha}x=g,\quad D^{k}_{\alpha}x(s)=D^{k}_{\alpha}f(s),\quad 0\leq
k\leq n.$

Note that the Cauchy function for $D^{n+1}_{\alpha}y=0$ is

$y(t,s)=\frac{1}{n!}\left(\frac{t^{\alpha}-s^{\alpha}}{\alpha}\right)^{n}.$

By the variation of constants formula,

$f(t)=u(t)+\frac{1}{n!}\int_{s}^{t}\left(\frac{t^{\alpha}-\tau^{\alpha}}{\alpha}\right)^{n}g(\tau)\tau^{\alpha-1}d\tau,$

where $u$ solves the initial value problem

(1.5) $D^{n+1}_{\alpha}u=0,\quad D^{m}_{\alpha}u(s)=D^{m}_{\alpha}f(s),\quad
0\leq m\leq n.$

To validate the claim that
$u(t)=\sum_{k=0}^{n}\frac{1}{k!}\left(\frac{t^{\alpha}-s^{\alpha}}{\alpha}\right)^{k}D^{k}_{\alpha}f(s)$,
set

$w(t):=\sum_{k=0}^{n}\frac{1}{k!}\left(\frac{t^{\alpha}-s^{\alpha}}{\alpha}\right)^{k}D^{k}_{\alpha}f(s).$

Then $D^{n+1}_{\alpha}w=0$, and we have that

$D^{m}_{\alpha}w(t)=\sum_{k=m}^{n}\frac{1}{(k-m)!}\left(\frac{t^{\alpha}-s^{\alpha}}{\alpha}\right)^{k-m}D^{k}_{\alpha}f(s).$

It follows that

$D^{m}_{\alpha}w(s)=\sum_{k=m}^{n}\frac{1}{(k-m)!}\left(\frac{s^{\alpha}-s^{\alpha}}{\alpha}\right)^{k-m}D^{k}_{\alpha}f(s)=D^{m}_{\alpha}f(s)$

for $0\leq m\leq n$. We consequently have that $w$ also solves (1.5), and thus
$u\equiv w$ by uniqueness. ∎

######  Corollary 1.3.

Let $\alpha\in(0,1]$ and $s,r\in[0,\infty)$ be fixed. For any $t\in[0,\infty)$
and any positive integer $n$,

$\frac{1}{n!}\left(\frac{t^{\alpha}-r^{\alpha}}{\alpha}\right)^{n}=\sum_{k=0}^{n}\frac{1}{k!(n-k)!}\left(\frac{t^{\alpha}-s^{\alpha}}{\alpha}\right)^{k}\left(\frac{s^{\alpha}-r^{\alpha}}{\alpha}\right)^{n-k}.$

###### Proof.

This follows immediately from the theorem if we take
$f(t)=\frac{1}{n!}\left(\frac{t^{\alpha}-r^{\alpha}}{\alpha}\right)^{n}$ in
Taylor’s formula. It can also be shown directly. ∎

##  2\. steffensen inequality

We begin this section with a definition of $\alpha$-fractional integrability.
The results in this and subsequent sections differ from those in [8].

######  Definition 2.1.

Let $\alpha\in(0,1]$ and $0\leq a<b$. A function $f:[a,b]\rightarrow\mbox{\Bbb
R}$ is $\alpha$-fractional integrable on $[a,b]$ if the integral

$\int_{a}^{b}f(t)d_{\alpha}t:=\int_{a}^{b}f(t)t^{\alpha-1}dt$

exists and is finite.

######  Lemma 2.1.

Let $\alpha\in(0,1]$ and $a,b\in\mbox{\Bbb R}$ with $0\leq a<b$. Let
$g:[a,b]\rightarrow[0,1]$ be an $\alpha$-fractional integrable function on
$[a,b]$, and define

(2.1)
$\ell:=\frac{\alpha(b-a)}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}g(t)d_{\alpha}t\in[0,b-a].$

Then

(2.2)
$\int_{b-\ell}^{b}1\;d_{\alpha}t\leq\int_{a}^{b}g(t)\;d_{\alpha}t\leq\int_{a}^{a+\ell}1\;d_{\alpha}t.$

###### Proof.

Since $g(t)\in[0,1]$ for all $t\in[a,b]$, $\ell$ given in (2.1) satisfies

$0\leq\ell=\frac{\alpha(b-a)}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}g(t)d_{\alpha}t\leq\frac{\alpha(b-a)}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}1\;d_{\alpha}t=\frac{\alpha(b-a)}{b^{\alpha}-a^{\alpha}}\frac{b^{\alpha}-a^{\alpha}}{\alpha}=b-a.$

As $\alpha\in(0,1]$ we have that $t^{\alpha-1}$ is a decreasing function on
$[a,b]$, or $(a,b]$ if $a=0$. Thus using the fact that
$d_{\alpha}t=t^{\alpha-1}dt$ we have the following inequalities, which are
average values, namely

$\frac{1}{\ell}\int_{b-\ell}^{b}1\;d_{\alpha}t\leq\frac{1}{b-a}\int_{a}^{b}1\;d_{\alpha}t\leq\frac{1}{\ell}\int_{a}^{a+\ell}1\;d_{\alpha}t.$

This implies that

$\int_{b-\ell}^{b}1\;d_{\alpha}t\leq\frac{\ell}{b-a}\int_{a}^{b}1\;d_{\alpha}t\leq\int_{a}^{a+\ell}1\;d_{\alpha}t,$

which leads to (2.2) via (2.1). ∎

######  Theorem 2.2 (Fractional Steffensen Inequality).

Let $\alpha\in(0,1]$ and $a,b\in\mbox{\Bbb R}$ with $0\leq a<b$. Let
$f:[a,b]\rightarrow[0,\infty)$ and $g:[a,b]\rightarrow[0,1]$ be
$\alpha$-fractional integrable functions on $[a,b]$, with $f$ decreasing. Then

(2.3)
$\int_{b-\ell}^{b}f(t)d_{\alpha}t\leq\int_{a}^{b}f(t)g(t)d_{\alpha}t\leq\int_{a}^{a+\ell}f(t)d_{\alpha}t,$

where $\ell$ is given by (2.1).

###### Proof.

We will prove only the case in (2.3) for the left inequality; the proof for
the right inequality is similar, and relies on (2.2).

By the definition of $\ell$ in (2.1) and the conditions on $g$, we know that
(2.2) holds. After subtracting within the left inequality of (2.3), we see
that

$\displaystyle\int_{a}^{b}f(t)g(t)d_{\alpha}t$ $\displaystyle-$
$\displaystyle\int_{b-\ell}^{b}f(t)d_{\alpha}t$ $\displaystyle=$
$\displaystyle\int_{a}^{b-\ell}f(t)g(t)d_{\alpha}t+\int_{b-\ell}^{b}f(t)g(t)d_{\alpha}t-\int_{b-\ell}^{b}f(t)d_{\alpha}t$
$\displaystyle=$
$\displaystyle\int_{a}^{b-\ell}f(t)g(t)d_{\alpha}t-\int_{b-\ell}^{b}f(t)(1-g(t))d_{\alpha}t$
$\displaystyle\geq$
$\displaystyle\int_{a}^{b-\ell}f(t)g(t)d_{\alpha}t-f(b-\ell)\int_{b-\ell}^{b}(1-g(t))d_{\alpha}t$
$\displaystyle=$
$\displaystyle\int_{a}^{b-\ell}f(t)g(t)d_{\alpha}t-f(b-\ell)\left[\int_{b-\ell}^{b}1\;d_{\alpha}t-\int_{b-\ell}^{b}g(t)d_{\alpha}t\right]$
$\displaystyle\stackrel{{\scriptstyle\eqref{gbds}}}{{\geq}}$
$\displaystyle\int_{a}^{b-\ell}f(t)g(t)d_{\alpha}t-f(b-\ell)\int_{a}^{b-\ell}g(t)d_{\alpha}t$
$\displaystyle=$
$\displaystyle\int_{a}^{b-\ell}\left(f(t)-f(b-\ell)\right)g(t)d_{\alpha}t\geq
0,$

since $f$ is decreasing and $g$ is nonnegative. Therefore the left-hand side
of (2.3) holds. ∎

######  Remark 2.1.

The requirement in Steffensen’s Theorem 2.2 that $f$ be non-negative is
essential. For example, let $a=0$, $b=1$, $\alpha=1/2\equiv g$, and
$f\equiv-1$. Then $\ell=1/2$, and

$\int_{b-\ell}^{b}f(t)d_{\alpha}t=-2+\sqrt{2}>-1=\int_{a}^{b}f(t)g(t)d_{\alpha}t,$

a contradiction of the left-hand side of (2.3).

##  3\. taylor remainder

Let $\alpha\in(0,1]$ and suppose $f$ is $n+1$ times $\alpha$-fractional
differentiable on $[0,\infty)$. Using Taylor’s Theorem, Theorem 1.2, we define
the remainder function by

$R_{-1,f}(\cdot,s):=f(s),$

and for $n>-1$,

$\displaystyle R_{n,f}(t,s)$ $\displaystyle:=$ $\displaystyle
f(s)-\sum_{k=0}^{n}\frac{D^{k}_{\alpha}f(t)}{k!}\left(\frac{s^{\alpha}-t^{\alpha}}{\alpha}\right)^{k}$
$\displaystyle=$
$\displaystyle\frac{1}{n!}\int_{t}^{s}\left(\frac{s^{\alpha}-\tau^{\alpha}}{\alpha}\right)^{n}D^{n+1}_{\alpha}f(\tau)d_{\alpha}\tau.$

######  Lemma 3.1.

Let $\alpha\in(0,1]$. The following identity involving $\alpha$-fractional
Taylor’s remainder holds:

$\int_{a}^{b}\frac{D^{n+1}_{\alpha}f(s)}{(n+1)!}\left(\frac{t^{\alpha}-s^{\alpha}}{\alpha}\right)^{n+1}d_{\alpha}s=\int_{a}^{t}R_{n,f}(a,s)d_{\alpha}s+\int_{t}^{b}R_{n,f}(b,s)d_{\alpha}s.$

###### Proof.

We proceed by mathematical induction on $n$. For $n=-1$,

$\int_{a}^{b}D^{0}_{\alpha}f(s)d_{\alpha}s=\int_{a}^{b}f(s)d_{\alpha}s=\int_{a}^{t}f(s)d_{\alpha}s+\int_{t}^{b}f(s)d_{\alpha}s.$

Assume the result holds for $n=k-1$:

$\int_{a}^{b}\frac{D^{k}_{\alpha}f(s)}{k!}\left(\frac{t^{\alpha}-s^{\alpha}}{\alpha}\right)^{k}d_{\alpha}s=\int_{a}^{t}R_{k-1,f}(a,s)d_{\alpha}s+\int_{t}^{b}R_{k-1,f}(b,s)d_{\alpha}s.$

Let $n=k$. Using integration by parts, we have

$\displaystyle\int_{a}^{b}\frac{D^{k+1}_{\alpha}f(s)}{(k+1)!}\left(\frac{t^{\alpha}-s^{\alpha}}{\alpha}\right)^{k+1}d_{\alpha}s$
$\displaystyle=$
$\displaystyle\frac{D^{k}_{\alpha}f(b)}{(k+1)!}\left(\frac{t^{\alpha}-b^{\alpha}}{\alpha}\right)^{k+1}$
$\displaystyle\;-\frac{D^{k}_{\alpha}f(a)}{(k+1)!}\left(\frac{t^{\alpha}-a^{\alpha}}{\alpha}\right)^{k+1}$
$\displaystyle\;+\int_{a}^{b}\frac{D^{k}_{\alpha}f(s)}{k!}\left(\frac{t^{\alpha}-s^{\alpha}}{\alpha}\right)^{k}d_{\alpha}s.$

By the induction assumption,

$\displaystyle\int_{a}^{b}\frac{D^{k+1}_{\alpha}f(s)}{(k+1)!}\left(\frac{t^{\alpha}-s^{\alpha}}{\alpha}\right)^{k+1}d_{\alpha}s$
$\displaystyle=$
$\displaystyle\int_{a}^{t}R_{k-1,f}(a,s)d_{\alpha}s+\int_{t}^{b}R_{k-1,f}(b,s)d_{\alpha}s$
$\displaystyle+\frac{D^{k}_{\alpha}f(b)}{(k+1)!}\left(\frac{t^{\alpha}-b^{\alpha}}{\alpha}\right)^{k+1}$
$\displaystyle-\frac{D^{k}_{\alpha}f(a)}{(k+1)!}\left(\frac{t^{\alpha}-a^{\alpha}}{\alpha}\right)^{k+1}$
$\displaystyle=$
$\displaystyle\int_{a}^{t}R_{k-1,f}(a,s)d_{\alpha}s+\int_{t}^{b}R_{k-1,f}(b,s)d_{\alpha}s$
$\displaystyle+\frac{D^{k}_{\alpha}f(b)}{k!}\int_{b}^{t}\left(\frac{s^{\alpha}-b^{\alpha}}{\alpha}\right)^{k}d_{\alpha}s$
$\displaystyle-\frac{D^{k}_{\alpha}f(a)}{k!}\int_{a}^{t}\left(\frac{s^{\alpha}-a^{\alpha}}{\alpha}\right)^{k}d_{\alpha}s$
$\displaystyle=$
$\displaystyle\int_{a}^{t}\left[R_{k-1,f}(a,s)-\frac{D^{k}_{\alpha}f(a)}{k!}\left(\frac{s^{\alpha}-a^{\alpha}}{\alpha}\right)^{k}\right]d_{\alpha}s$
$\displaystyle+\int_{t}^{b}\left[R_{k-1,f}(b,s)-\frac{D^{k}_{\alpha}f(b)}{k!}\left(\frac{s^{\alpha}-b^{\alpha}}{\alpha}\right)^{k}\right]d_{\alpha}s$
$\displaystyle=$
$\displaystyle\int_{a}^{t}R_{k,f}(a,s)d_{\alpha}s+\int_{t}^{b}R_{k,f}(b,s)d_{\alpha}s.$

This completes the proof. ∎

######  Corollary 3.2.

Let $\alpha\in(0,1]$. For $n\geq-1$,

$\displaystyle\int_{a}^{b}\frac{D^{n+1}_{\alpha}f(s)}{(n+1)!}\left(\frac{a^{\alpha}-s^{\alpha}}{\alpha}\right)^{n+1}d_{\alpha}s=\int_{a}^{b}R_{n,f}(b,s)d_{\alpha}s,$
$\displaystyle\int_{a}^{b}\frac{D^{n+1}_{\alpha}f(s)}{(n+1)!}\left(\frac{b^{\alpha}-s^{\alpha}}{\alpha}\right)^{n+1}d_{\alpha}s=\int_{a}^{b}R_{n,f}(a,s)d_{\alpha}s.$

##  4\. applications of the steffensen inequality

Let $\alpha\in(0,1]$. In the following we adapt to the $\alpha$-fractional
setting some results from [5] by applying the fractional Steffensen
inequality, Theorem 2.2.

######  Theorem 4.1.

Let $\alpha\in(0,1]$ and $f:[a,b]\rightarrow\mbox{\Bbb R}$ be an $n+1$ times
$\alpha$-fractional differentiable function such that $D^{n+1}_{\alpha}f$ is
increasing and $D^{n}_{\alpha}f$ is decreasing on $[a,b]$. If

$\ell:=\frac{b-a}{n+2},$

then

$\displaystyle D^{n}_{\alpha}f(a+\ell)-D^{n}_{\alpha}f(a)$ $\displaystyle\leq$
$\displaystyle(n+1)!\left(\frac{\alpha}{b^{\alpha}-a^{\alpha}}\right)^{n+1}\int_{a}^{b}R_{n,f}(a,s)d_{\alpha}s$
$\displaystyle\leq$ $\displaystyle
D^{n}_{\alpha}f(b)-D^{n}_{\alpha}f(b-\ell).$

###### Proof.

Let $F:=-D^{n+1}_{\alpha}f$. Because $D^{n}_{\alpha}f$ is decreasing,
$D^{n+1}_{\alpha}f\leq 0$, so that $F\geq 0$ and decreasing on $[a,b]$. Define

$g(t):=\left(\frac{b^{\alpha}-t^{\alpha}}{b^{\alpha}-a^{\alpha}}\right)^{n+1}\in[0,1],\quad
t\in[a,b],\quad n\geq-1.$

Note that $F,g$ satisfy the assumptions of Steffensen’s inequality, Theorem
2.2; using (2.1),

$\ell=\frac{\alpha(b-a)}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}g(t)d_{\alpha}t=\frac{b-a}{n+2},$

and

$-\int_{b-\ell}^{b}D^{n+1}_{\alpha}f(t)d_{\alpha}t\leq-\int_{a}^{b}D^{n+1}_{\alpha}f(t)\left(\frac{b^{\alpha}-t^{\alpha}}{b^{\alpha}-a^{\alpha}}\right)^{n+1}d_{\alpha}t\leq-\int_{a}^{a+\ell}D^{n+1}_{\alpha}f(t)d_{\alpha}t.$

By Corollary 3.2 this simplifies to

$D^{n}_{\alpha}f(t)|_{t=a}^{a+\ell}\leq(n+1)!\left(\frac{\alpha}{b^{\alpha}-a^{\alpha}}\right)^{n+1}\int_{a}^{b}R_{n,f}(a,t)d_{\alpha}t\leq
D^{n}_{\alpha}f(t)|^{b}_{t=b-\ell}.$

This completes the proof. ∎

The following corollary is the first Hermite-Hadamard inequality, derived from
Theorem 4.1 with $n=0$.

######  Corollary 4.2 (Hermite-Hadamard Inequality I).

Let $\alpha\in(0,1]$ and $f:[a,b]\rightarrow\mbox{\Bbb R}$ be an
$\alpha$-fractional differentiable function such that $D_{\alpha}f$ is
increasing and $f$ is decreasing on $[a,b]$. Then

$\displaystyle
f\left(\frac{a+b}{2}\right)\leq\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(s)d_{\alpha}s\leq
f(b)+f(a)-f\left(\frac{a+b}{2}\right).$

######  Theorem 4.3.

Let $\alpha\in(0,1]$ and $f:[a,b]\rightarrow\mbox{\Bbb R}$ be an $n+1$ times
$\alpha$-fractional differentiable function such that

$m\leq D^{n+1}_{\alpha}f\leq M$

on $[a,b]$ for some real numbers $m<M$. Then

$\displaystyle\frac{m}{(n+2)!}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+2}+\frac{M-m}{(n+2)!}\left(\frac{b^{\alpha}-(b-\ell)^{\alpha}}{\alpha}\right)^{n+2}\leq\int_{a}^{b}R_{n,f}(a,t)d_{\alpha}t$
(4.1)
$\displaystyle\leq\frac{M}{(n+2)!}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+2}+\frac{m-M}{(n+2)!}\left(\frac{b^{\alpha}-(a+\ell)^{\alpha}}{\alpha}\right)^{n+2},$

where $\ell$ is given by

$\ell=\frac{\alpha(b-a)}{\left(b^{\alpha}-a^{\alpha}\right)(M-m)}\left(D^{n}_{\alpha}f(b)-D^{n}_{\alpha}f(a)-m\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)\right)$

###### Proof.

Let

$\displaystyle F(t)$ $\displaystyle:=$
$\displaystyle\frac{1}{(n+1)!}\left(\frac{b^{\alpha}-t^{\alpha}}{\alpha}\right)^{n+1},$
$\displaystyle k(t)$ $\displaystyle:=$
$\displaystyle\frac{1}{M-m}\left(f(t)-\frac{m}{(n+1)!}\left(\frac{t^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+1}\right),$
$\displaystyle G(t)$ $\displaystyle:=$ $\displaystyle
D^{n+1}_{\alpha}k(t)=\frac{1}{M-m}\left(D^{n+1}_{\alpha}f(t)-m\right)\in[0,1].$

Observe that $F$ is nonnegative and decreasing, and

$\int_{a}^{b}G(t)d_{\alpha}t=\frac{1}{M-m}\left(D^{n}_{\alpha}f(b)-D^{n}_{\alpha}f(a)-m\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)\right).$

Since $F,G$ satisfy the hypotheses of Theorem 2.2, we compute the various
integrals given in (2.3), after using (2.1) to set

$\ell=\frac{\alpha(b-a)}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}G(t)d_{\alpha}t.$

We have

$\int_{b-\ell}^{b}F(t)d_{\alpha}t=\int_{b-\ell}^{b}\frac{1}{(n+1)!}\left(\frac{b^{\alpha}-t^{\alpha}}{\alpha}\right)^{n+1}d_{\alpha}t=\frac{1}{(n+2)!}\left(\frac{b^{\alpha}-(b-\ell)^{\alpha}}{\alpha}\right)^{n+2},$

and

$\int_{a}^{a+\ell}F(t)d_{\alpha}t=\frac{1}{(n+2)!}\left[\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+2}-\left(\frac{b^{\alpha}-(a+\ell)^{\alpha}}{\alpha}\right)^{n+2}\right].$

Moreover, using Corollary 3.2, we have

$\displaystyle\int_{a}^{b}F(t)G(t)d_{\alpha}t$ $\displaystyle=$
$\displaystyle\frac{1}{(M-m)(n+1)!}\int_{a}^{b}\left(\frac{b^{\alpha}-t^{\alpha}}{\alpha}\right)^{n+1}\left(D^{n+1}_{\alpha}f(t)-m\right)d_{\alpha}t$
$\displaystyle=$
$\displaystyle\frac{1}{M-m}\int_{a}^{b}R_{n,f}(a,t)d_{\alpha}t-\frac{m}{(M-m)(n+2)!}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+2}.$

Using Steffensen’s inequality (2.3) and some rearranging, we obtain (4.1). ∎

######  Corollary 4.4.

Let $\alpha\in(0,1]$ and $f:[a,b]\rightarrow\mbox{\Bbb R}$ be an
$\alpha$-fractional differentiable function such that

$m\leq D_{\alpha}f\leq M$

on $[a,b]$ for some real numbers $m<M$. Then

$\displaystyle\frac{m}{2}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{2}+\frac{M-m}{2}\left(\frac{b^{\alpha}-(b-\ell)^{\alpha}}{\alpha}\right)^{2}\leq\int_{a}^{b}f(t)d_{\alpha}t-f(a)\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)$
(4.2)
$\displaystyle\leq\frac{M}{2}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{2}+\frac{m-M}{2}\left(\frac{b^{\alpha}-(a+\ell)^{\alpha}}{\alpha}\right)^{2},$

where $\ell$ is given by

$\ell=\frac{\alpha(b-a)}{\left(b^{\alpha}-a^{\alpha}\right)(M-m)}\left(f(b)-f(a)-m\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)\right)$

###### Proof.

Use the previous theorem with $n=0$, and Corollary 3.2. ∎

##  5\. applications of the $\breve{\text{C}}$eby$\breve{\text{s}}$ev
inequality

Let $\alpha\in(0,1]$. We begin with
$\breve{\text{C}}$eby$\breve{\text{s}}$ev’s inequality for $\alpha$-fractional
integrals, then apply it to obtain a Hermite-Hadamard-type inequality.

######  Theorem 5.1 ($\breve{\text{C}}$eby$\breve{\text{s}}$ev Inequality).

Let $f$ and $g$ be both increasing or both decreasing in $[a,b]$, and let
$\alpha\in(0,1]$. Then

$\int_{a}^{b}f(t)g(t)d_{\alpha}t\geq\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(t)d_{\alpha}t\int_{a}^{b}g(t)d_{\alpha}t.$

If one of the functions is increasing and the other is decreasing, then the
above inequality is reversed.

###### Proof.

The proof is very similar to the classical case with $\alpha=1$. ∎

The following is an application of $\breve{\text{C}}$eby$\breve{\text{s}}$ev’s
inequality, which extends a similar result in [5] for $q$-calculus to this
$\alpha$-fractional case.

######  Theorem 5.2.

Let $\alpha\in(0,1]$. Assume that $D^{n+1}_{\alpha}f$ is monotonic on $[a,b]$.
If $D^{n+1}_{\alpha}f$ is increasing, then

$\displaystyle 0$ $\displaystyle\geq$
$\displaystyle\int_{a}^{b}R_{n,f}(a,t)d_{\alpha}t-\left(\frac{D^{n}_{\alpha}f(b)-D^{n}_{\alpha}f(a)}{(n+2)!}\right)\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+1}$
$\displaystyle\geq$
$\displaystyle\left(\frac{D^{n+1}_{\alpha}f(a)-D^{n+1}_{\alpha}f(b)}{(n+2)!}\right)\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+2}.$

If $D^{n+1}_{\alpha}f$ is decreasing, then the inequalities are reversed.

###### Proof.

The situation where $D^{n+1}_{\alpha}f$ is decreasing is analogous to that of
$D^{n+1}_{\alpha}f$ increasing. Thus, assume $D^{n+1}_{\alpha}f$ is increasing
and set

$F(t):=D^{n+1}_{\alpha}f(t),\quad
G(t):=\frac{1}{(n+1)!}\left(\frac{b^{\alpha}-t^{\alpha}}{\alpha}\right)^{n+1}.$

Then $F$ is increasing by assumption, and $G$ is decreasing, so that by
$\breve{\text{C}}$eby$\breve{\text{s}}$ev’s inequality,

$\int_{a}^{b}F(t)G(t)d_{\alpha}t\leq\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}F(t)d_{\alpha}t\int_{a}^{b}G(t)d_{\alpha}t.$

By Corollary 3.2,

$\int_{a}^{b}F(t)G(t)d_{\alpha}t=\int_{a}^{b}\frac{D^{n+1}_{\alpha}f(t)}{(n+1)!}\left(\frac{b^{\alpha}-t^{\alpha}}{\alpha}\right)^{n+1}d_{\alpha}t=\int_{a}^{b}R_{n,f}(a,t)d_{\alpha}t.$

We also have

$\int_{a}^{b}F(t)d_{\alpha}t=D^{n}_{\alpha}f(b)-D^{n}_{\alpha}f(a),\quad\int_{a}^{b}G(t)d_{\alpha}t=\frac{1}{(n+2)!}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+2}.$

Thus $\breve{\text{C}}$eby$\breve{\text{s}}$ev’s inequality implies

$\int_{a}^{b}R_{n,f}(a,t)d_{\alpha}t\leq\frac{\alpha}{b^{\alpha}-a^{\alpha}}\left(D^{n}_{\alpha}f(b)-D^{n}_{\alpha}f(a)\right)\frac{1}{(n+2)!}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+2},$

which subtracts to the left side of the inequality. Since $D^{n+1}_{\alpha}f$
is increasing on $[a,b]$,

$\displaystyle\frac{D^{n+1}_{\alpha}f(a)}{(n+2)!}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+2}$
$\displaystyle\leq$
$\displaystyle\left(\frac{D^{n}_{\alpha}f(b)-D^{n}_{\alpha}f(a)}{(n+2)!}\right)\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+1}$
$\displaystyle\leq$
$\displaystyle\frac{D^{n+1}_{\alpha}f(b)}{(n+2)!}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+2},$

and we have

$\displaystyle\int_{a}^{b}R_{n,f}(a,t)d_{\alpha}t-\left(\frac{D^{n}_{\alpha}f(b)-D^{n}_{\alpha}f(a)}{(n+2)!}\right)\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+1}$
$\displaystyle\geq\int_{a}^{b}R_{n,f}(a,t)d_{\alpha}t-\frac{D^{n+1}_{\alpha}f(b)}{(n+2)!}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+2}.$

Now Corollary 3.2 and $D^{n+1}_{\alpha}f$ is increasing imply that

$\displaystyle\int_{a}^{b}\frac{D^{n+1}_{\alpha}f(b)}{(n+1)!}\left(\frac{b^{\alpha}-t^{\alpha}}{\alpha}\right)^{n+1}d_{\alpha}t$
$\displaystyle\geq$ $\displaystyle\int_{a}^{b}R_{n,f}(a,t)d_{\alpha}t$
$\displaystyle\geq$
$\displaystyle\int_{a}^{b}\frac{D^{n+1}_{\alpha}f(a)}{(n+1)!}\left(\frac{b^{\alpha}-t^{\alpha}}{\alpha}\right)^{n+1}d_{\alpha}t,$

which simplifies to

$\frac{D^{n+1}_{\alpha}f(b)}{(n+2)!}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+2}\geq\int_{a}^{b}R_{n,f}(a,t)d_{\alpha}t\geq\frac{D^{n+1}_{\alpha}f(a)}{(n+2)!}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)^{n+2}.$

This, together with the earlier lines give the right side of the inequality. ∎

Compare the following corollary with Corollary 4.2.

######  Corollary 5.3 (Hermite-Hadamard Inequality II).

Let $\alpha\in(0,1]$. If $D_{\alpha}f$ is increasing on $[a,b]$, then

(5.1)
$\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(t)d_{\alpha}t\leq\frac{f(b)+f(a)}{2}.$

If $D_{\alpha}f$ is decreasing on $[a,b]$, then the inequalities are reversed.

##  6\. Ostrowski Inequality

In this section we prove Ostrowski’s $\alpha$-fractional inequality using a
Montgomery identity. For more on Ostrowski’s inequalities, see [1] and the
references therein.

######  Lemma 6.1 (Montgomery Identity).

Let $a,b,s,t\in\mbox{\Bbb R}$ with $0\leq a<b$, and let
$f:[a,b]\rightarrow\mbox{\Bbb R}$ be $\alpha$-fractional differentiable for
$\alpha\in(0,1]$. Then

(6.1)
$f(t)=\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(s)d_{\alpha}s+\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}p(t,s)D_{\alpha}f(s)d_{\alpha}s$

where

(6.2) $p(t,s):=\begin{cases}\frac{s^{\alpha}-a^{\alpha}}{\alpha}&:a\leq
s<t,\\\ \frac{s^{\alpha}-b^{\alpha}}{\alpha}&:t\leq s\leq b.\end{cases}$

###### Proof.

Integrating by parts, we have

$\int_{a}^{t}\left(\frac{s^{\alpha}-a^{\alpha}}{\alpha}\right)D_{\alpha}f(s)d_{\alpha}s=\frac{t^{\alpha}-a^{\alpha}}{\alpha}f(t)-\int_{a}^{t}f(s)d_{\alpha}s$

and

$\int_{t}^{b}\left(\frac{s^{\alpha}-b^{\alpha}}{\alpha}\right)D_{\alpha}f(s)d_{\alpha}s=\frac{b^{\alpha}-t^{\alpha}}{\alpha}f(t)-\int_{t}^{b}f(s)d_{\alpha}s.$

Adding and solving for $f$ yields the result. ∎

######  Theorem 6.2 (Ostrowski Inequality).

Let $a,b,s,t\in\mbox{\Bbb R}$ with $0\leq a<b$, and let
$f:[a,b]\rightarrow\mbox{\Bbb R}$ be $\alpha$-fractional differentiable for
$\alpha\in(0,1]$. Then

(6.3)
$\left|f(t)-\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(t)d_{\alpha}t\right|\leq\frac{M}{2\alpha\left(b^{\alpha}-a^{\alpha}\right)}\left[\left(t^{\alpha}-a^{\alpha}\right)^{2}+\left(b^{\alpha}-t^{\alpha}\right)^{2}\right],$

where

$M:=\sup_{t\in(a,b)}\left|D_{\alpha}f(t)\right|.$

This inequality is sharp in the sense that the right-hand side of (6.3) cannot
be replaced by a smaller one.

###### Proof.

Using Lemma 6.1 with $p(t,s)$ defined in 6.2 we see that

$\displaystyle\left|f(t)-\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(s)d_{\alpha}s\right|$
$\displaystyle=$
$\displaystyle\left|\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}p(t,s)D_{\alpha}f(s)d_{\alpha}s\right|$
$\displaystyle\leq$
$\displaystyle\frac{M\alpha}{b^{\alpha}-a^{\alpha}}\left(\int_{a}^{t}\left|\frac{s^{\alpha}-a^{\alpha}}{\alpha}\right|d_{\alpha}s+\int_{t}^{b}\left|\frac{s^{\alpha}-b^{\alpha}}{\alpha}\right|d_{\alpha}s\right)$
$\displaystyle=$
$\displaystyle\frac{M\alpha}{b^{\alpha}-a^{\alpha}}\left(\int_{a}^{t}\left(\frac{s^{\alpha}-a^{\alpha}}{\alpha}\right)d_{\alpha}s+\int_{t}^{b}\left(\frac{b^{\alpha}-s^{\alpha}}{\alpha}\right)d_{\alpha}s\right)$
$\displaystyle=$
$\displaystyle\frac{M\alpha}{b^{\alpha}-a^{\alpha}}\left(\frac{1}{2}\left(\frac{s^{\alpha}-a^{\alpha}}{\alpha}\right)^{2}\Big{|}_{a}^{t}-\frac{1}{2}\left(\frac{b^{\alpha}-s^{\alpha}}{\alpha}\right)^{2}\Big{|}_{t}^{b}\right)$
$\displaystyle=$
$\displaystyle\frac{M}{2\alpha\left(b^{\alpha}-a^{\alpha}\right)}\left[\left(t^{\alpha}-a^{\alpha}\right)^{2}+\left(b^{\alpha}-t^{\alpha}\right)^{2}\right].$

Now $p(t,a)=0$, so the smallest value attaining the supremum in $M$ is greater
than $a$. To prove the sharpness of this inequality, let
$f(t)=t^{\alpha}/\alpha$, $a=t_{1}$, $b=t_{2}=t$. It follows that
$D_{\alpha}f(t)=1$ and $M=1$. Examining the right-hand side of (6.3) we get

$\frac{M}{2\alpha\left(b^{\alpha}-a^{\alpha}\right)}\left[\left(t^{\alpha}-a^{\alpha}\right)^{2}+\left(b^{\alpha}-t^{\alpha}\right)^{2}\right]=\frac{\left(t_{2}^{\alpha}-t_{1}^{\alpha}\right)^{2}}{2\alpha\left(t_{2}^{\alpha}-t_{1}^{\alpha}\right)}=\frac{t_{2}^{\alpha}-t_{1}^{\alpha}}{2\alpha}.$

Starting with the left-hand side of (6.3), we have

$\displaystyle\left|f(t)-\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(t)d_{\alpha}t\right|$
$\displaystyle=$
$\displaystyle\left|\frac{t^{\alpha}}{\alpha}-\frac{\alpha}{t_{2}^{\alpha}-t_{1}^{\alpha}}\int_{t_{1}}^{t_{2}}\frac{t^{\alpha}}{\alpha}d_{\alpha}t\right|$
$\displaystyle=$
$\displaystyle\left|\frac{t^{\alpha}}{\alpha}-\left(\frac{\alpha}{t_{2}^{\alpha}-t_{1}^{\alpha}}\right)\left(\frac{t^{2\alpha}}{2\alpha^{2}}\right)\Big{|}_{t_{1}}^{t_{2}}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{t^{\alpha}}{\alpha}-\left(\frac{1}{t_{2}^{\alpha}-t_{1}^{\alpha}}\right)\left(\frac{t_{2}^{2\alpha}-t_{1}^{2\alpha}}{2\alpha}\right)\right|$
$\displaystyle=$
$\displaystyle\left|\frac{t^{\alpha}}{\alpha}-\left(\frac{t_{2}^{\alpha}+t_{1}^{\alpha}}{2\alpha}\right)\right|$
$\displaystyle=$ $\displaystyle\frac{t_{2}^{\alpha}-t_{1}^{\alpha}}{2\alpha}.$

Therefore by the squeeze theorem the sharpness of Ostrowski’s inequality is
shown. ∎

##  7\. Grüss Inequality

In this section we prove the Grüss inequality, which relies on Jensen’s
inequality. Our approach is similar to that taken by [2].

######  Theorem 7.1 (Jensen Inequality).

Let $\alpha\in(0,1]$ and $a,b,x,y\in[0,\infty)$. If $w:\mbox{\Bbb
R}\rightarrow\mbox{\Bbb R}$ and $g:\mbox{\Bbb R}\rightarrow(x,y)$ are
nonnegative, continuous functions with $\int_{a}^{b}w(t)d_{\alpha}t>0$, and
$F:(x,y)\rightarrow\mbox{\Bbb R}$ is continuous and convex, then

$F\left(\frac{\int_{a}^{b}w(t)g(t)d_{\alpha}t}{\int_{a}^{b}w(t)d_{\alpha}t}\right)\leq\frac{\int_{a}^{b}w(t)F(g(t))d_{\alpha}t}{\int_{a}^{b}w(t)d_{\alpha}t}.$

###### Proof.

The proof is the same as those found in Bohner and Peterson [3, Theorem 6.17]
and Rudin [9, Theorem 3.3] and thus is omitted. ∎

######  Theorem 7.2 (Grüss Inequality).

Let $a,b,s\in[0,\infty)$, and let $f,g:[a,b]\rightarrow\mbox{\Bbb R}$ be
continuous functions. Then for $\alpha\in(0,1]$ and

(7.1) $m_{1}\leq f(t)\leq M_{1},\qquad m_{2}\leq g(t)\leq M_{2},$

we have

$\displaystyle\left|\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(t)g(t)d_{\alpha}t-\left(\frac{\alpha}{b^{\alpha}-a^{\alpha}}\right)^{2}\int_{a}^{b}f(t)d_{\alpha}t\int_{a}^{b}g(t)d_{\alpha}t\right|$
$\displaystyle\leq\frac{1}{4}(M_{1}-m_{1})(M_{2}-m_{2}).$

###### Proof.

Initially we consider an easier case, namely where $f=g$ and

$\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(t)d_{\alpha}t=0.$

If we define

$v(t):=\frac{f(t)-m_{1}}{M_{1}-m_{1}}\in[0,1],$

then $f(t)=m_{1}+(M_{1}-m_{1})v(t)$. Since

$\int_{a}^{b}v^{2}(t)d_{\alpha}t\leq\int_{a}^{b}v(t)d_{\alpha}t=\frac{-m_{1}(b^{\alpha}-a^{\alpha})}{\alpha(M_{1}-m_{1})},$

we have

$\displaystyle I(f,f)$ $\displaystyle:=$
$\displaystyle\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f^{2}(t)d_{\alpha}t-\left(\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(t)d_{\alpha}t\right)^{2}$
$\displaystyle=$
$\displaystyle\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}\left[m_{1}+(M_{1}-m_{1})v(t)\right]^{2}(t)d_{\alpha}t$
$\displaystyle\leq$ $\displaystyle-
m_{1}M_{1}=\frac{1}{4}\left[(M_{1}-m_{1})^{2}-(M_{1}+m_{1})^{2}\right]$
$\displaystyle\leq$ $\displaystyle\frac{1}{4}(M_{1}-m_{1})^{2}.$

Now consider the case

$r:=\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(t)d_{\alpha}t\not=0,$

where $r\in\mbox{\Bbb R}$. If we take $h(t):=f(t)-r$, then
$h(t)\in[m_{1}-r,M_{1}-r]$ and

$\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}h(t)d_{\alpha}t=\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}(f(t)-r)d_{\alpha}t=r-\frac{r\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}d_{\alpha}t=0.$

Consequently $h$ satisfies the earlier assumptions and so

$I(h,h)\leq\frac{1}{4}\left[M_{1}-r-(m_{1}-r)\right]^{2}=\frac{1}{4}\left(M_{1}-m_{1}\right)^{2}.$

Additionally we have

$I(h,h)=\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}(f(t)-r)^{2}d_{\alpha}t=-r^{2}+\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f^{2}(t)d_{\alpha}t=I(f,f).$

As a result,

$I(f,f)=I(h,h)\leq\frac{1}{4}\left(M_{1}-m_{1}\right)^{2}.$

Let us now turn to the case involving general functions $f$ and $g$ under
assumptions (7.1). Using

$\displaystyle I(f,g)$ $\displaystyle:=$
$\displaystyle\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(t)g(t)d_{\alpha}t-\left(\frac{\alpha}{b^{\alpha}-a^{\alpha}}\right)^{2}\int_{a}^{b}f(t)d_{\alpha}t\int_{a}^{b}g(t)d_{\alpha}t$

and the earlier cases, one can easily finish the proof as in the case with
$\alpha=1$. See [2] for complete details to mimic. ∎

######  Corollary 7.3.

Let $\alpha\in(0,1]$, $a,b,s,t\in[0,\infty)$ and $f:[a,b]\rightarrow\mbox{\Bbb
R}$ be $\alpha$-fractional differentiable. If $D_{\alpha}f$ is continuous and

$m\leq D_{\alpha}f(t)\leq M,\quad t\in[a,b],$

then

$\displaystyle\left|f(t)-\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(s)d_{\alpha}s-\left[\frac{2t^{\alpha}-a^{\alpha}-b^{\alpha}}{2(b^{\alpha}-a^{\alpha})}\right]\left[f(b)-f(a)\right]\right|$
(7.2)
$\displaystyle\leq\frac{1}{4}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)(M-m).$

for all $t\in[a,b]$.

###### Proof.

Using Lemma 6.1 Montgomery’s identity we have

(7.3)
$f(t)-\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(s)d_{\alpha}s=\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}p(t,s)D_{\alpha}f(s)d_{\alpha}s$

for all $t\in[a,b]$, where $p(t,s)$ is given in (6.2). Now for all
$t,s\in[a,b]$, we see that

$\frac{t^{\alpha}-b^{\alpha}}{\alpha}\leq
p(t,s)\leq\frac{t^{\alpha}-a^{\alpha}}{\alpha}.$

Applying Theorem 7.2 Grüss’ inequality to the mappings $p(t,\cdot)$ and
$D_{\alpha}f$, we obtain

$\displaystyle\left|\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}p(t,s)D_{\alpha}f(s)d_{\alpha}s-\left(\frac{\alpha}{b^{\alpha}-a^{\alpha}}\right)^{2}\int_{a}^{b}p(t,s)d_{\alpha}s\int_{a}^{b}D_{\alpha}f(s)d_{\alpha}s\right|$
(7.4)
$\displaystyle\leq\frac{1}{4}\left(\frac{t^{\alpha}-a^{\alpha}}{\alpha}-\frac{t^{\alpha}-b^{\alpha}}{\alpha}\right)(M-m)=\frac{1}{4}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)(M-m).$

Computing the integrals involved, we obtain

$\displaystyle\left(\frac{\alpha}{b^{\alpha}-a^{\alpha}}\right)^{2}\int_{a}^{b}p(t,s)d_{\alpha}s=\frac{2t^{\alpha}-a^{\alpha}-b^{\alpha}}{2(b^{\alpha}-a^{\alpha})}$

and

$\displaystyle\int_{a}^{b}D_{\alpha}f(s)d_{\alpha}s=f(b)-f(a),$

so that (7.2) holds, after using (7.3) and (7.4). ∎

Compare the following corollary with Corollary 4.2 and Corollary 5.3.

######  Corollary 7.4 (Hermite-Hadamard III).

Let $\alpha\in(0,1]$, $a,b,s,t\in[0,\infty)$ and $f:[a,b]\rightarrow\mbox{\Bbb
R}$ be $\alpha$-fractional differentiable. If $D_{\alpha}f$ is continuous and

$m\leq D_{\alpha}f(t)\leq M,\quad t\in[a,b],$

then

$\left|\frac{f(b)+f(a)}{2}-\frac{\alpha}{b^{\alpha}-a^{\alpha}}\int_{a}^{b}f(s)d_{\alpha}s\right|\leq\frac{1}{4}\left(\frac{b^{\alpha}-a^{\alpha}}{\alpha}\right)(M-m).$

for all $t\in[a,b]$.

###### Proof.

Take $t=b$ in the previous corollary. ∎

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