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224 | $3.114\times 10^{-15}$
225 | $2.786\times 10^{-15}$
226 | $2.481\times 10^{-15}$
227 | $2.197\times 10^{-15}$
228 | $1.932\times 10^{-15}$
229 | $1.687\times 10^{-15}$
230 | $1.460\times 10^{-15}$
231 | $1.250\times 10^{-15}$
232 | $1.059\times 10^{-15}$
233 | $8.865\times 10^{-16}$
234 | $7.324\times 10^{-16}$
235 | $5.974\times 10^{-16}$
236 | $4.816\times 10^{-16}$
237 | $3.843\times 10^{-16}$
238 | $3.042\times 10^{-16}$
239 | $2.392\times 10^{-16}$

240 | $1.870\times 10^{-16}$
---|---
241 | $1.456\times 10^{-16}$
242 | $1.129\times 10^{-16}$
243 | $8.731\times 10^{-17}$
244 | $6.756\times 10^{-17}$
245 | $5.253\times 10^{-17}$
246 | $4.126\times 10^{-17}$
247 | $3.288\times 10^{-17}$
248 | $2.664\times 10^{-17}$
249 | $2.197\times 10^{-17}$
250 | $1.841\times 10^{-17}$
251 | $1.565\times 10^{-17}$
252 | $1.347\times 10^{-17}$
253 | $1.172\times 10^{-17}$
254 | $1.029\times 10^{-17}$
255 | $9.099\times 10^{-18}$
256 | $8.101\times 10^{-18}$
257 | $7.253\times 10^{-18}$
258 | $6.523\times 10^{-18}$
259 | $5.890\times 10^{-18}$
260 | $5.336\times 10^{-18}$
261 | $4.847\times 10^{-18}$
262 | $4.413\times 10^{-18}$
263 | $4.025\times 10^{-18}$
264 | $3.676\times 10^{-18}$
265 | $3.361\times 10^{-18}$
266 | $3.075\times 10^{-18}$
267 | $2.814\times 10^{-18}$
268 | $2.576\times 10^{-18}$
269 | $2.357\times 10^{-18}$
270 | $2.155\times 10^{-18}$
271 | $1.969\times 10^{-18}$
272 | $1.797\times 10^{-18}$
273 | $1.637\times 10^{-18}$
274 | $1.488\times 10^{-18}$
275 | $1.349\times 10^{-18}$
276 | $1.219\times 10^{-18}$
277 | $1.097\times 10^{-18}$
278 | $9.839\times 10^{-19}$
279 | $8.774\times 10^{-19}$
280 | $7.777\times 10^{-19}$
281 | $6.844\times 10^{-19}$
282 | $5.971\times 10^{-19}$
283 | $5.157\times 10^{-19}$
284 | $4.402\times 10^{-19}$
285 | $3.706\times 10^{-19}$
286 | $3.073\times 10^{-19}$
287 | $2.506\times 10^{-19}$
288 | $2.009\times 10^{-19}$
289 | $1.585\times 10^{-19}$
290 | $1.235\times 10^{-19}$
291 | $9.538\times 10^{-20}$
292 | $7.328\times 10^{-20}$
293 | $5.610\times 10^{-20}$
294 | $4.280\times 10^{-20}$
295 | $3.250\times 10^{-20}$
296 | $2.451\times 10^{-20}$
297 | $1.835\times 10^{-20}$
298 | $1.367\times 10^{-20}$
299 | $1.019\times 10^{-20}$

300 | $7.682\times 10^{-21}$
---|---
301 | $5.902\times 10^{-21}$
302 | $4.645\times 10^{-21}$
303 | $3.746\times 10^{-21}$
304 | $3.088\times 10^{-21}$
305 | $2.595\times 10^{-21}$
306 | $2.215\times 10^{-21}$
307 | $1.916\times 10^{-21}$
308 | $1.675\times 10^{-21}$
309 | $1.478\times 10^{-21}$
310 | $1.314\times 10^{-21}$
311 | $1.175\times 10^{-21}$
312 | $1.057\times 10^{-21}$
313 | $9.546\times 10^{-22}$
314 | $8.656\times 10^{-22}$
315 | $7.873\times 10^{-22}$
316 | $7.181\times 10^{-22}$
317 | $6.563\times 10^{-22}$
318 | $6.009\times 10^{-22}$
319 | $5.510\times 10^{-22}$
320 | $5.058\times 10^{-22}$
321 | $4.646\times 10^{-22}$
322 | $4.270\times 10^{-22}$
323 | $3.925\times 10^{-22}$
324 | $3.607\times 10^{-22}$
325 | $3.314\times 10^{-22}$
326 | $3.042\times 10^{-22}$
327 | $2.790\times 10^{-22}$
328 | $2.555\times 10^{-22}$
329 | $2.336\times 10^{-22}$
330 | $2.132\times 10^{-22}$
331 | $1.940\times 10^{-22}$
332 | $1.760\times 10^{-22}$
333 | $1.591\times 10^{-22}$
334 | $1.432\times 10^{-22}$
335 | $1.282\times 10^{-22}$
336 | $1.141\times 10^{-22}$
337 | $1.008\times 10^{-22}$
338 | $8.823\times 10^{-23}$
339 | $7.644\times 10^{-23}$
340 | $6.536\times 10^{-23}$
341 | $5.503\times 10^{-23}$
342 | $4.545\times 10^{-23}$
343 | $3.670\times 10^{-23}$
344 | $2.889\times 10^{-23}$
345 | $2.214\times 10^{-23}$
346 | $1.659\times 10^{-23}$
347 | $1.225\times 10^{-23}$
348 | $9.012\times 10^{-24}$
349 | $6.647\times 10^{-24}$
350 | $4.925\times 10^{-24}$
351 | $3.656\times 10^{-24}$
352 | $2.705\times 10^{-24}$
353 | $1.980\times 10^{-24}$
354 | $1.424\times 10^{-24}$
355 | $1.000\times 10^{-24}$
356 | $6.867\times 10^{-25}$
357 | $4.688\times 10^{-25}$
358 | $3.272\times 10^{-25}$
359 | $2.381\times 10^{-25}$

360 | $1.813\times 10^{-25}$
---|---
361 | $1.434\times 10^{-25}$
362 | $1.171\times 10^{-25}$
363 | $9.786\times 10^{-26}$
364 | $8.336\times 10^{-26}$
365 | $7.207\times 10^{-26}$
366 | $6.305\times 10^{-26}$
367 | $5.570\times 10^{-26}$
368 | $4.960\times 10^{-26}$
369 | $4.446\times 10^{-26}$
370 | $4.007\times 10^{-26}$
371 | $3.628\times 10^{-26}$
372 | $3.297\times 10^{-26}$
373 | $3.007\times 10^{-26}$
374 | $2.750\times 10^{-26}$
375 | $2.520\times 10^{-26}$
376 | $2.314\times 10^{-26}$
377 | $2.128\times 10^{-26}$
378 | $1.960\times 10^{-26}$
379 | $1.806\times 10^{-26}$
380 | $1.666\times 10^{-26}$
381 | $1.537\times 10^{-26}$
382 | $1.418\times 10^{-26}$
383 | $1.308\times 10^{-26}$
384 | $1.207\times 10^{-26}$
385 | $1.112\times 10^{-26}$
386 | $1.024\times 10^{-26}$
387 | $9.416\times 10^{-27}$
388 | $8.645\times 10^{-27}$
389 | $7.922\times 10^{-27}$
390 | $7.243\times 10^{-27}$
391 | $6.603\times 10^{-27}$
392 | $5.999\times 10^{-27}$
393 | $5.430\times 10^{-27}$
394 | $4.891\times 10^{-27}$
395 | $4.380\times 10^{-27}$
396 | $3.897\times 10^{-27}$
397 | $3.438\times 10^{-27}$
398 | $3.002\times 10^{-27}$
399 | $2.589\times 10^{-27}$
400 | $2.196\times 10^{-27}$
401 | $1.825\times 10^{-27}$
402 | $1.475\times 10^{-27}$
403 | $1.148\times 10^{-27}$
404 | $8.493\times 10^{-28}$
405 | $5.878\times 10^{-28}$
406 | $3.797\times 10^{-28}$
407 | $2.373\times 10^{-28}$
408 | $1.516\times 10^{-28}$
409 | $1.014\times 10^{-28}$
410 | $7.052\times 10^{-29}$
411 | $5.024\times 10^{-29}$
412 | $3.610\times 10^{-29}$
413 | $2.575\times 10^{-29}$
414 | $1.789\times 10^{-29}$
415 | $1.172\times 10^{-29}$
416 | $6.766\times 10^{-30}$
417 | $2.700\times 10^{-30}$

Table S3: Numerical values of region-region logarithmic negativity
$\mathcal{N}_{A|B}(\sigma)$ for $m=3\times 10^{-3}/d$ presented in Fig. 3.

d = 20
---
$\tilde{r}$ | $\mathcal{N}(\sigma_{h})_{A|B}$
0 | $4.818\times 10^{-1}$
1 | $4.862\times 10^{-1}$
2 | $5.027\times 10^{-1}$
3 | $5.444\times 10^{-1}$
4 | $5.852\times 10^{-1}$
5 | $6.224\times 10^{-1}$
6 | $6.562\times 10^{-1}$
7 | $1.953\times 10^{-1}$
8 | $1.733\times 10^{-1}$
9 | $1.522\times 10^{-1}$
10 | $1.250\times 10^{-1}$
11 | $4.682\times 10^{-2}$
12 | $2.720\times 10^{-2}$
13 | $1.697\times 10^{-2}$
14 | $1.144\times 10^{-2}$
15 | $8.260\times 10^{-3}$
16 | $6.366\times 10^{-3}$
17 | $5.167\times 10^{-3}$
18 | $4.262\times 10^{-3}$
19 | $3.578\times 10^{-3}$
20 | $3.097\times 10^{-3}$
21 | $2.783\times 10^{-3}$
22 | $2.541\times 10^{-3}$
23 | $2.414\times 10^{-3}$
24 | $2.339\times 10^{-3}$
25 | $2.303\times 10^{-3}$
26 | $2.315\times 10^{-3}$
27 | $2.356\times 10^{-3}$
28 | $2.416\times 10^{-3}$
29 | $2.488\times 10^{-3}$
30 | $2.569\times 10^{-3}$
31 | $2.649\times 10^{-3}$
32 | $2.720\times 10^{-3}$
33 | $2.768\times 10^{-3}$
34 | $2.782\times 10^{-3}$
35 | $2.746\times 10^{-3}$
36 | $2.631\times 10^{-3}$
37 | $2.483\times 10^{-3}$
38 | $2.255\times 10^{-3}$
39 | $1.947\times 10^{-3}$
40 | $1.571\times 10^{-3}$
41 | $1.152\times 10^{-3}$
42 | $7.358\times 10^{-4}$
43 | $4.036\times 10^{-4}$
44 | $2.179\times 10^{-4}$
45 | $1.333\times 10^{-4}$
46 | $9.177\times 10^{-5}$
47 | $6.856\times 10^{-5}$
48 | $5.454\times 10^{-5}$
49 | $4.523\times 10^{-5}$
50 | $3.871\times 10^{-5}$
51 | $3.401\times 10^{-5}$
52 | $3.056\times 10^{-5}$
53 | $2.802\times 10^{-5}$
54 | $2.615\times 10^{-5}$
55 | $2.474\times 10^{-5}$
56 | $2.378\times 10^{-5}$
57 | $2.313\times 10^{-5}$
58 | $2.274\times 10^{-5}$
59 | $2.255\times 10^{-5}$

60 | $2.262\times 10^{-5}$
---|---
61 | $2.290\times 10^{-5}$
62 | $2.334\times 10^{-5}$
63 | $2.395\times 10^{-5}$
64 | $2.470\times 10^{-5}$
65 | $2.555\times 10^{-5}$
66 | $2.644\times 10^{-5}$
67 | $2.726\times 10^{-5}$
68 | $2.790\times 10^{-5}$
69 | $2.823\times 10^{-5}$
70 | $2.815\times 10^{-5}$
71 | $2.759\times 10^{-5}$
72 | $2.640\times 10^{-5}$
73 | $2.463\times 10^{-5}$
74 | $2.229\times 10^{-5}$
75 | $1.937\times 10^{-5}$
76 | $1.598\times 10^{-5}$
77 | $1.225\times 10^{-5}$
78 | $8.371\times 10^{-6}$
79 | $4.698\times 10^{-6}$
80 | $2.065\times 10^{-6}$
81 | $9.659\times 10^{-7}$
82 | $5.701\times 10^{-7}$
83 | $3.910\times 10^{-7}$
84 | $2.933\times 10^{-7}$
85 | $1.883\times 10^{-7}$
86 | $1.594\times 10^{-7}$
87 | $1.392\times 10^{-7}$
88 | $1.249\times 10^{-7}$
89 | $1.148\times 10^{-7}$
90 | $1.075\times 10^{-7}$
91 | $1.025\times 10^{-7}$
92 | $9.904\times 10^{-8}$
93 | $9.684\times 10^{-8}$
94 | $9.553\times 10^{-8}$
95 | $9.476\times 10^{-8}$
96 | $9.426\times 10^{-8}$
97 | $9.390\times 10^{-8}$
98 | $9.382\times 10^{-8}$
99 | $9.411\times 10^{-8}$
100 | $9.479\times 10^{-8}$
101 | $9.599\times 10^{-8}$
102 | $9.790\times 10^{-8}$
103 | $1.006\times 10^{-7}$
104 | $1.042\times 10^{-7}$
105 | $1.084\times 10^{-7}$
106 | $1.129\times 10^{-7}$
107 | $1.171\times 10^{-7}$
108 | $1.205\times 10^{-7}$
109 | $1.226\times 10^{-7}$
110 | $1.227\times 10^{-7}$
111 | $1.206\times 10^{-7}$
112 | $1.159\times 10^{-7}$
113 | $1.085\times 10^{-7}$
114 | $9.860\times 10^{-8}$
115 | $8.632\times 10^{-8}$
116 | $7.209\times 10^{-8}$
117 | $5.646\times 10^{-8}$
118 | $4.024\times 10^{-8}$
119 | $2.433\times 10^{-8}$

120 | $1.049\times 10^{-8}$
---|---
121 | $3.449\times 10^{-9}$
122 | $1.625\times 10^{-9}$
123 | $1.005\times 10^{-9}$
124 | $7.117\times 10^{-10}$
125 | $5.454\times 10^{-10}$
126 | $4.412\times 10^{-10}$
127 | $3.718\times 10^{-10}$
128 | $3.239\times 10^{-10}$
129 | $2.903\times 10^{-10}$
130 | $2.667\times 10^{-10}$
131 | $2.504\times 10^{-10}$
132 | $2.396\times 10^{-10}$
133 | $2.330\times 10^{-10}$
134 | $2.296\times 10^{-10}$
135 | $2.286\times 10^{-10}$
136 | $2.291\times 10^{-10}$
137 | $2.304\times 10^{-10}$
138 | $2.314\times 10^{-10}$
139 | $2.317\times 10^{-10}$
140 | $2.309\times 10^{-10}$
141 | $2.285\times 10^{-10}$
142 | $2.091\times 10^{-10}$
143 | $2.060\times 10^{-10}$
144 | $2.040\times 10^{-10}$
145 | $2.044\times 10^{-10}$
146 | $2.080\times 10^{-10}$
147 | $2.151\times 10^{-10}$
148 | $2.254\times 10^{-10}$
149 | $2.382\times 10^{-10}$
150 | $2.522\times 10^{-10}$
151 | $2.660\times 10^{-10}$
152 | $2.782\times 10^{-10}$
153 | $2.873\times 10^{-10}$
154 | $2.919\times 10^{-10}$
155 | $2.912\times 10^{-10}$
156 | $2.844\times 10^{-10}$
157 | $2.713\times 10^{-10}$
158 | $2.519\times 10^{-10}$
159 | $2.267\times 10^{-10}$
160 | $1.965\times 10^{-10}$
161 | $1.624\times 10^{-10}$
162 | $1.258\times 10^{-10}$
163 | $8.813\times 10^{-11}$
164 | $5.128\times 10^{-11}$
165 | $1.900\times 10^{-11}$
166 | $4.610\times 10^{-12}$
167 | $2.033\times 10^{-12}$
168 | $1.239\times 10^{-12}$
169 | $8.700\times 10^{-13}$
170 | $6.615\times 10^{-13}$
171 | $5.304\times 10^{-13}$
172 | $4.427\times 10^{-13}$
173 | $3.818\times 10^{-13}$
174 | $3.388\times 10^{-13}$
175 | $3.084\times 10^{-13}$
176 | $2.872\times 10^{-13}$
177 | $2.731\times 10^{-13}$
178 | $2.648\times 10^{-13}$
179 | $2.610\times 10^{-13}$

180 | $2.611\times 10^{-13}$
---|---
181 | $2.643\times 10^{-13}$
182 | $2.697\times 10^{-13}$
183 | $2.762\times 10^{-13}$
184 | $2.822\times 10^{-13}$
185 | $2.859\times 10^{-13}$
186 | $2.853\times 10^{-13}$
187 | $2.787\times 10^{-13}$
188 | $2.661\times 10^{-13}$
189 | $2.496\times 10^{-13}$
190 | $2.329\times 10^{-13}$
191 | $2.197\times 10^{-13}$
192 | $2.126\times 10^{-13}$
193 | $2.123\times 10^{-13}$
194 | $2.190\times 10^{-13}$
195 | $2.318\times 10^{-13}$
196 | $2.495\times 10^{-13}$
197 | $2.706\times 10^{-13}$
198 | $2.933\times 10^{-13}$
199 | $3.156\times 10^{-13}$
200 | $3.356\times 10^{-13}$
201 | $3.515\times 10^{-13}$
202 | $3.616\times 10^{-13}$
203 | $3.648\times 10^{-13}$
204 | $3.602\times 10^{-13}$
205 | $3.474\times 10^{-13}$
206 | $3.266\times 10^{-13}$
207 | $2.984\times 10^{-13}$
208 | $2.638\times 10^{-13}$
209 | $2.242\times 10^{-13}$
210 | $1.813\times 10^{-13}$
211 | $1.369\times 10^{-13}$
212 | $9.274\times 10^{-14}$
213 | $5.057\times 10^{-14}$
214 | $1.436\times 10^{-14}$
215 | $2.509\times 10^{-15}$
216 | $1.125\times 10^{-15}$
217 | $6.987\times 10^{-16}$
218 | $4.971\times 10^{-16}$
219 | $3.817\times 10^{-16}$
220 | $3.084\times 10^{-16}$
221 | $2.590\times 10^{-16}$
222 | $2.244\times 10^{-16}$
223 | $2.000\times 10^{-16}$
224 | $1.827\times 10^{-16}$
225 | $1.708\times 10^{-16}$
226 | $1.633\times 10^{-16}$
227 | $1.594\times 10^{-16}$
228 | $1.586\times 10^{-16}$
229 | $1.606\times 10^{-16}$
230 | $1.652\times 10^{-16}$
231 | $1.721\times 10^{-16}$
232 | $1.810\times 10^{-16}$
233 | $1.909\times 10^{-16}$
234 | $2.006\times 10^{-16}$
235 | $2.078\times 10^{-16}$
236 | $2.096\times 10^{-16}$
237 | $2.029\times 10^{-16}$
238 | $1.861\times 10^{-16}$
239 | $1.614\times 10^{-16}$

240 | $1.351\times 10^{-16}$
---|---
241 | $1.139\times 10^{-16}$
242 | $1.006\times 10^{-16}$
243 | $9.483\times 10^{-17}$
244 | $9.515\times 10^{-17}$
245 | $1.005\times 10^{-16}$
246 | $1.101\times 10^{-16}$
247 | $1.232\times 10^{-16}$
248 | $1.389\times 10^{-16}$
249 | $1.562\times 10^{-16}$
250 | $1.739\times 10^{-16}$
251 | $1.910\times 10^{-16}$
252 | $2.062\times 10^{-16}$
253 | $2.185\times 10^{-16}$
254 | $2.268\times 10^{-16}$
255 | $2.305\times 10^{-16}$
256 | $2.290\times 10^{-16}$
257 | $2.221\times 10^{-16}$
258 | $2.099\times 10^{-16}$
259 | $1.930\times 10^{-16}$
260 | $1.720\times 10^{-16}$
261 | $1.480\times 10^{-16}$
262 | $1.220\times 10^{-16}$
263 | $9.527\times 10^{-17}$
264 | $6.880\times 10^{-17}$
265 | $4.349\times 10^{-17}$
266 | $2.008\times 10^{-17}$
267 | $2.482\times 10^{-18}$
268 | $4.809\times 10^{-19}$
269 | $2.475\times 10^{-19}$
270 | $1.628\times 10^{-19}$
271 | $1.198\times 10^{-19}$
272 | $9.413\times 10^{-20}$
273 | $7.742\times 10^{-20}$
274 | $6.596\times 10^{-20}$
275 | $5.787\times 10^{-20}$
276 | $5.213\times 10^{-20}$
277 | $4.811\times 10^{-20}$
278 | $4.545\times 10^{-20}$
279 | $4.390\times 10^{-20}$
280 | $4.335\times 10^{-20}$
281 | $4.372\times 10^{-20}$
282 | $4.500\times 10^{-20}$
283 | $4.721\times 10^{-20}$
284 | $5.041\times 10^{-20}$
285 | $5.462\times 10^{-20}$
286 | $5.982\times 10^{-20}$
287 | $6.576\times 10^{-20}$
288 | $7.180\times 10^{-20}$
289 | $7.671\times 10^{-20}$
290 | $7.855\times 10^{-20}$
291 | $7.503\times 10^{-20}$
292 | $6.456\times 10^{-20}$
293 | $4.816\times 10^{-20}$
294 | $3.176\times 10^{-20}$
295 | $2.161\times 10^{-20}$
296 | $1.699\times 10^{-20}$
297 | $1.534\times 10^{-20}$
298 | $1.542\times 10^{-20}$
299 | $1.680\times 10^{-20}$

300 | $1.932\times 10^{-20}$
---|---
301 | $2.287\times 10^{-20}$
302 | $2.734\times 10^{-20}$
303 | $3.255\times 10^{-20}$
304 | $3.834\times 10^{-20}$
305 | $4.446\times 10^{-20}$
306 | $5.067\times 10^{-20}$
307 | $5.664\times 10^{-20}$
308 | $6.205\times 10^{-20}$
309 | $6.655\times 10^{-20}$
310 | $6.980\times 10^{-20}$
311 | $7.155\times 10^{-20}$
312 | $7.161\times 10^{-20}$
313 | $6.990\times 10^{-20}$
314 | $6.646\times 10^{-20}$
315 | $6.146\times 10^{-20}$
316 | $5.516\times 10^{-20}$
317 | $4.791\times 10^{-20}$
318 | $4.011\times 10^{-20}$
319 | $3.215\times 10^{-20}$
320 | $2.436\times 10^{-20}$
321 | $1.700\times 10^{-20}$
322 | $1.024\times 10^{-20}$
323 | $4.186\times 10^{-21}$
324 | $1.796\times 10^{-22}$
325 | $3.953\times 10^{-23}$
326 | $2.149\times 10^{-23}$
327 | $1.456\times 10^{-23}$
328 | $1.094\times 10^{-23}$
329 | $8.744\times 10^{-24}$
330 | $7.301\times 10^{-24}$
331 | $6.305\times 10^{-24}$
332 | $5.602\times 10^{-24}$
333 | $5.105\times 10^{-24}$
334 | $4.763\times 10^{-24}$
335 | $4.548\times 10^{-24}$
336 | $4.441\times 10^{-24}$
337 | $4.437\times 10^{-24}$
338 | $4.534\times 10^{-24}$
339 | $4.742\times 10^{-24}$
340 | $5.075\times 10^{-24}$
341 | $5.560\times 10^{-24}$
342 | $6.229\times 10^{-24}$
343 | $7.124\times 10^{-24}$
344 | $8.275\times 10^{-24}$
345 | $9.658\times 10^{-24}$
346 | $1.111\times 10^{-23}$
347 | $1.226\times 10^{-23}$
348 | $1.252\times 10^{-23}$
349 | $1.131\times 10^{-23}$
350 | $8.285\times 10^{-24}$
351 | $3.912\times 10^{-24}$
352 | $1.327\times 10^{-24}$
353 | $7.591\times 10^{-25}$
354 | $6.104\times 10^{-25}$
355 | $5.952\times 10^{-25}$
356 | $6.676\times 10^{-25}$
357 | $8.299\times 10^{-25}$
358 | $1.092\times 10^{-24}$
359 | $1.460\times 10^{-24}$

360 | $1.939\times 10^{-24}$
---|---
361 | $2.537\times 10^{-24}$
362 | $3.275\times 10^{-24}$
363 | $4.177\times 10^{-24}$
364 | $5.274\times 10^{-24}$
365 | $6.595\times 10^{-24}$
366 | $8.162\times 10^{-24}$
367 | $9.978\times 10^{-24}$
368 | $1.201\times 10^{-23}$
369 | $1.421\times 10^{-23}$
370 | $1.647\times 10^{-23}$
371 | $1.869\times 10^{-23}$
372 | $2.074\times 10^{-23}$
373 | $2.251\times 10^{-23}$
374 | $2.390\times 10^{-23}$
375 | $2.480\times 10^{-23}$
376 | $2.516\times 10^{-23}$
377 | $2.491\times 10^{-23}$
378 | $2.402\times 10^{-23}$
379 | $2.250\times 10^{-23}$
380 | $2.037\times 10^{-23}$
381 | $1.773\times 10^{-23}$
382 | $1.475\times 10^{-23}$
383 | $1.166\times 10^{-23}$
384 | $8.733\times 10^{-24}$
385 | $6.184\times 10^{-24}$
386 | $4.125\times 10^{-24}$
387 | $2.536\times 10^{-24}$
388 | $1.337\times 10^{-24}$
389 | $4.339\times 10^{-25}$
390 | $2.655\times 10^{-27}$
391 | $7.780\times 10^{-28}$
392 | $4.614\times 10^{-28}$
393 | $3.325\times 10^{-28}$
394 | $2.637\times 10^{-28}$
395 | $2.220\times 10^{-28}$
396 | $1.950\times 10^{-28}$
397 | $1.773\times 10^{-28}$
398 | $1.660\times 10^{-28}$
399 | $1.598\times 10^{-28}$
400 | $1.583\times 10^{-28}$
401 | $1.618\times 10^{-28}$
402 | $1.714\times 10^{-28}$
403 | $1.902\times 10^{-28}$
404 | $2.237\times 10^{-28}$
405 | $2.836\times 10^{-28}$
406 | $3.892\times 10^{-28}$
407 | $5.582\times 10^{-28}$
408 | $7.878\times 10^{-28}$
409 | $1.060\times 10^{-27}$
410 | $1.361\times 10^{-27}$
411 | $1.682\times 10^{-27}$
412 | $2.019\times 10^{-27}$
413 | $2.371\times 10^{-27}$
414 | $2.737\times 10^{-27}$
415 | $3.116\times 10^{-27}$
416 | $3.509\times 10^{-27}$
417 | $3.916\times 10^{-27}$

Table S4: Numerical values of region-region SOE negativity in the halo
$\mathcal{N}_{A|B}(\sigma_{h})$ for $m=3\times 10^{-3}/d$ presented in Fig. 3.


# Some properties of geodesic $(\alpha,E)$-preinvex functions on a Riemannian
manifold

Absos Ali Shaikh1, Chandan Kumar Mondal2, Ravi P Agarwal3
1Department of Mathematics,
University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
2School of sciences,
Netaji Subhas Open University,
Durgapur Campus, Durgapur-713214,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
3Department of Mathematics,
Texas A&M University-Kingsville, Kingsville,
Texas 78363-8202,
USA<EMAIL_ADDRESS>
###### Abstract.

In this article, we have introduced the concept of geodesic $(\alpha,E)$-invex
set and by using this concept the notion of geodesic $(\alpha,E)$-preinvex
functions and geodesic $(\alpha,E)$-invex functions are developed on a
Riemannian manifold. Moreover, several properties and results are deduced
within aforesaid functions. An example is also constructed to illustrate the
definition of geodesic $(\alpha,E)$-invex set. We have also established an
important relation between geodesic $(\alpha,E)$-preinvex function and
geodesic $(\alpha,E)$-invex function in a complete Riemannian manifold.

††footnotetext:  ∗ Corresponding author.
$\mathbf{2010}$ Mathematics Subject Classification: 53C22, 58E10, 53B20.
Key words and phrases: Geodesic $(\alpha,E)$-Invex sets, $(\alpha,E)$-Invex
functions, Geodesic $(\alpha,E)$-Preinvex functions, Riemannian manifolds

##  1\. Introduction

Convex sets and convex functions play an important and significant role in the
theory of nonlinear programming and optimization. Since, the notion of
convexity has great impact in real world problems, various authors are
developing the new concept of convexity in order to extend the result to the
larger class of optimization. Hanson in 1981, made a significant step by
introducing the concept of invexity. Hanson’s work stimulates further
development of the role and applications of invexity in mathematical
programming and other branches of pure and applied mathematics. Later, the
role of preinvex functions and invex sets in optimization theory, variational
inequalities and equilibrium problems were studied by Jeyakumar [5], Weir and
Mond [18] and various authors. In 1999, Younees [16] generalized the concept
of convexity by placing an operator $E$ in the same domain and named it as
$E$-convex set and $E$-convex function. However, Yang [15] proved that some of
the results given by Youness are not correct. Again the concept of strong
$E$-convexity and semi-strong $E$-convexity was defined in [17].

Over the past few years, many results in the theory of nonlinear analysis and
optimization on Riemannian manifolds have been extended from the Euclidean
space. Convex functions in Riemannian manifolds, were studied by many authors
see [3, 9, 12, 13, 19]. Udrişte [14] and Rapcsak [11] developed the concept of
geodesic convexity in Riemannian manifold. Pini [10] introduced the concept of
Riemannian invexity, while Mititelu [8] investigated its generalization. In
2012, Iqbal et al. [4] established the notion of geodesic E-convex set and
functions and also they have discussed their properties and results. Barani et
al. [2] introduced the concepts of geodesic invex set and geodesic preinvex
functions on Riemannian manifolds with respect to particular maps. In 2012,
Agarwal et al. [1] generalized the notion of invexity and developed the
concept of $\alpha$-invex sets and $\alpha$-preinvex functions in Riemannian
manifold. Agian, Kumari and Jayswal [7] introduced the notion of geodesic
E-preinvex function and geodesic semi E-preinvex function on Riemannian
manifold and investigated some of its properties. Motivating by the work of
Agarwal et al. and Kumari and Jayswal, we have developed the concept of
$(\alpha,E)$-invex sets and $(\alpha,E)$-preinvex functions in a complete
Riemannian manifold and investigated some of its properties.

The paper is structured as follows: Section 2 deals with some well known facts
of Riemannian manifolds, geodesic convexity and geodesic invexity. In the next
section, we have defined geodesic $(\alpha,E)$-invex set in a complete
Riemannian manifold. And by using this definition we have developed the
concept of geodesic $(\alpha,E)$-preinvex and geodesic $(\alpha,E)$-invex
functions. In the last section, we have deduced a relation between geodesic
$(\alpha,E)$-preinvex and geodesic $(\alpha,E)$-invex function,( see Theorem
4.2), which is the main result of this paper. Also we have deduced some
properties of $(\alpha,E)$-preinvex and geodesic $(\alpha,E)$-invex functions.

##  2\. Notations and Preliminaries

In this section we have recalled some basic concept of a Riemannian manifold
$(M,g)$, which is necessary throughout this paper (for reference see [6]). The
length $L(\gamma)$ of the curve $\gamma:[a,b]\rightarrow M$ is given by

$\displaystyle L(\gamma)$ $\displaystyle=$
$\displaystyle\int_{a}^{b}\sqrt{g_{\gamma(t)}(\dot{\gamma}(t),\dot{\gamma}(t))}\
dt$ $\displaystyle=$
$\displaystyle\int_{a}^{b}\left\lVert\dot{\gamma}(t)\right\rVert dt.$

The curve $\gamma$ is said to be a geodesic if its velocity vector
$\gamma^{\prime}(t)$ is parallel along $\gamma(t)$, i.e.,
$\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)=0\ \forall t\in[a,b]$, where $\nabla$
is the Riemannian connection of $g$. For any point $p\in M$ and the unique
geodesic $\gamma_{u}$ with $\gamma(0)=p$ and $\dot{\gamma}_{u}(0)=u$, the
exponential map $exp_{p}:V_{p}\rightarrow M$ is defined by

$exp_{p}(u)=\gamma_{u}(1),$

where and $V_{p}$ is a collection of vectors of $T_{p}M$ such that for each
element $u\in V_{p}$, the geodesic with initial tangent vector $u$ is defined
on $[0,1]$. It can be easily seen that the norm of a tangent vector is
constant for a geodesic $\gamma$. A smooth vector field is a smooth function
$X:M\rightarrow TM$ such that $\pi\circ X=id_{M}$, where $\pi:TM\rightarrow M$
is the projection map, i.e, in each point of the manifold $M$ we smoothly
choose a tangent vector. The gradient of a function $f\in C^{\infty}(M)$ at
the point $p\in M$ is defined by $\nabla f(p)=g^{ij}\frac{\partial f}{\partial
x_{j}}\frac{\partial}{\partial x_{i}}\mid_{p},$ where
$\\{\frac{\partial}{\partial x_{i}}\\}_{i=1}^{n}|_{p}$ is an orthonormal
coordinate system for $T_{p}M$.

######  Definition 2.1.

[14] A real valued function $f$ on $M$ is called convex if

$f\circ\gamma(t)\leq(1-t)f\circ\gamma(0)+tf\circ\gamma(1)\quad\forall
t\in[0,1],$

for every geodesic $\gamma:[0,1]\rightarrow M$.

######  Definition 2.2.

[2] Let $M$ be a Riemannian manifold and $\eta:M\times M\rightarrow TM$ be a
function such that for every $x,y\in M$, $\eta(x,y)\in T_{y}M$. A nonempty
subset $S$ of $M$ is said to be geodesic invex with respect to $\eta$ if for
every $x,y\in S$ there exists exactly one geodesic
$\gamma_{x,y}:[0,1]\rightarrow M$ such that

$\gamma_{x,y}(0)=y,\ \gamma^{\prime}_{x,y}(0)=\eta(x,y),\ \gamma_{x,y}(t)\in
S,\quad\forall t\in[0,1].$

######  Definition 2.3.

[2] Let $S$ be an open set of the Riemannian manifold $M$ which is geodesic
invex set with respect to $\eta:M\times M\rightarrow TM$. A differentiable
function $f:S\rightarrow\mathbb{R}$ is said to be $\eta$-invex on $S$ if the
following condition holds

$f(x)-f(y)\geq df_{y}(\eta(x,y)),\ \forall x,y\in S.$

$f$ is called geodesic $\eta$-preinvex if for every $x,y\in S$,

$f(\gamma_{x,y}(t))\leq tf(x)+(1-t)f(y),\ \forall t\in[0,1].$

##  3\. $(\alpha,E)$-invex set and $(\alpha,E)$-preinvex function

Suppose $E:M\rightarrow M$ is a function and $\alpha:M\times
M\rightarrow\mathbb{R}-\\{0\\}$ is a bifunction. We denote the complete
Riemannian manifold by the notation $M$.

######  Definition 3.1.

Let $\eta:M\times M\rightarrow TM$ be a function and $\alpha:M\times
M\rightarrow\mathbb{R}-\\{0\\}$ be a bifunction such that
$\alpha(x,y)\eta(x,y)\in T_{y}M$ for every $x,y\in M$. A subset $S\neq\phi$ of
$M$ is said to be a geodesic $(\alpha,E)$-invex set with respect to $\eta$ and
$\alpha$ if for every $x,y\in S$, there exists exactly one geodesic
$\gamma_{E(x),E(y)}:[0,1]\rightarrow M$ such that

$\gamma_{E(x),E(y)}(0)=E(y),\
\gamma^{\prime}_{E(x),E(y)}(0)=\alpha(E(x),E(y))\eta(E(x),E(y)),\
\gamma_{E(x),E(y)}(t)\in S,\quad\forall t\in[0,1].$

If $\alpha(p,q)=1$ for all $p,q\in M$ and $E$ is the identity map, then
geodesic $(\alpha,E)-$invex set becomes geodesic invex set [2]. Geodesic
$(\alpha,E)-$invex set reduces to geodesic $\alpha$-invex set [1] if $E(p)=p$
for all $p\in M$. If only $\alpha\equiv 1$, then geodesic $(\alpha,E)-$invex
set becomes geodesic $E$-invex set [7].

######  Definition 3.2.

Let $S$ be an open subset of $M$ which is geodesic $(\alpha,E)$-invex set with
respect to $\eta:M\times M\rightarrow TM$ and $\alpha:M\times
M\rightarrow\mathbb{R}-\\{0\\}$. A function $f:S\rightarrow\mathbb{R}$ is said
to be geodesic $(\alpha,E)-$preinvex if

$f(\gamma_{E(x),E(y)}(t))\leq tf(E(x))+(1-t)f(E(y)),\quad\forall x,y\in
S\text{ and }t\in[0,1],$

where $\gamma_{E(x),E(y)}$ is the unique geodesic.

######  Definition 3.3.

Let $S$ be an open subset of $M$ which is geodesic $(\alpha,E)$-invex set with
respect to $\eta:M\times M\rightarrow TM$ and $\alpha:M\times
M\rightarrow\mathbb{R}-\\{0\\}$. A differentiable function
$f:S\rightarrow\mathbb{R}$ is said to be geodesic $(\alpha,E)-$invex if

$f(E(x))-f(E(y))\geq df_{E(y)}(\alpha(E(x),E(y))\eta(E(x),E(y))),\quad\forall
x,y\in S.$

######  Example 3.1.

Let $M$ be a Cartan-Hadamard manifold and $x_{0},y_{0}\in M$ such that
$x_{0}\neq y_{0}$. Consider two open ball $B(x_{0},r_{1})$ and
$B(y_{0},r_{2})$ of radius $r_{1}$ and $r_{2}$ respectively, such that
$B(x_{0},r_{1})\cap B(y_{0},r_{2})=\phi$ for some
$0<r_{1},r_{2}<\frac{1}{2}d(x_{0},y_{0})$. Now take

$A=B(x_{0},r_{1})\cup B(y_{0},r_{2}).$

Then it is obvious that $A$ is not geodesic convex. Now define the functions
$E:M\rightarrow M$ and $\eta:M\times M\rightarrow M$ such that

$\eta(x,y)=\begin{cases}exp_{y}^{-1}x&x,y\in B(x_{0},r_{1})\text{ or }x,y\in
B(y_{0},r_{2})\\\ 0_{y}&\text{ otherwise.}\end{cases}$
$E(x)=\\{y\in\gamma_{x,y}:d(x_{0},y)=r_{1}/2\\}\text{ for all }x\in A,$

where $\gamma_{x,y}$ denotes the geodesic joining $x$ and $y$ whose existence
is ensured in Cartan-Hadamard manifolds.
For every $x,y\in M$, consider a bifunction $\alpha:M\times
M\rightarrow\mathbb{R}-\\{0\\}$ and $\gamma_{E(x),E(y)}:[0,1]\rightarrow M$
defined by

$\gamma_{E(x),E(y)}(t)=exp_{E(y)}(t\alpha(E(x),E(y))\eta(E(x),E(y))),\text{
for all }t\in[0,1].$

Then,

$\gamma_{E(x),E(y)}(0)=E(y),\
\gamma^{\prime}_{E(x),E(y)}(0)=\alpha(E(x),E(y))\eta(E(x),E(y)).$

Now simple calculation shows that $A$ is a geodesic $(\alpha,E)$-invex set.

##  4\. Main Results

######  Proposition 4.1.

Suppose $S$ is an open subset of $M$ which is geodesic $(\alpha,E)$-invex set
with respect to $\eta:M\times M\rightarrow TM$ and $\alpha:M\times
M\rightarrow\mathbb{R}-\\{0\\}$. Suppose $f:S\rightarrow\mathbb{R}$ is a
geodesic $(\alpha,E)$-preinvex function, then every lower section of $f$
defined by

$S_{\lambda}:=\\{x\in S:f(x)\leq\lambda\\},\quad\lambda\in\mathbb{R},$

is a geodesic $(\alpha,E)$-invex set with respect to $\eta$ and $\alpha$.

###### Proof.

Let $x,y\in S_{\lambda}$. Since $S$ is a geodesic $(\alpha,E)$-invex set with
respect to $\eta$ and $\alpha$, there exists exactly one geodesic
$\gamma_{E(x),E(y)}:[0,1]\rightarrow M$ such that

$\gamma_{E(x),E(y)}(0)=E(y),\
\gamma^{\prime}_{E(x),E(y)}(0)=\alpha(E(x),E(y))\eta(E(x),E(y)),\
\gamma_{E(x),E(y)}(t)\in S,\quad\forall t\in[0,1].$

Now by geodesic $(\alpha,E)$-preinvexity of $f$ we have

$\displaystyle f(\gamma_{E(x),E(y)}(t))$ $\displaystyle\leq$ $\displaystyle
tf(E(x))+(1-t)f(E(y))$ $\displaystyle\leq$ $\displaystyle
t\lambda+(1-t)\lambda=\lambda.$

Hence $\gamma_{E(x),E(y)}(t)\in S_{\lambda}\ \forall t\in[0,1]$. Therefore,
$S_{\lambda}$ is a geodesic $(\alpha,E)$-invex set with respect to $\eta$ and
$\alpha$. ∎

######  Theorem 4.2.

Let $S\subset M$ be a geodesic $(\alpha,E)$-invex set with respect to $\eta$
and $\alpha$. If a function $f:S\rightarrow\mathbb{R}$ is differentiable and
geodesic $(\alpha,E)$-preinvex on $S$, then $f$ is a geodesic
$(\alpha,E)$-invex function on $S$.

###### Proof.

Since $S$ is a geodesic $(\alpha,E)$-invex set with respect to $\eta$ and
$\alpha$, there exists exactly one geodesic
$\gamma_{E(x),E(y)}:[0,1]\rightarrow M$ such that

$\gamma_{E(x),E(y)}(0)=E(y),\
\gamma^{\prime}_{E(x),E(y)}(0)=\alpha(E(x),E(y))\eta(E(x),E(y)),\
\gamma_{E(x),E(y)}(t)\in S,\quad\forall t\in[0,1].$

Again $f$ is geodesic $(\alpha,E)$-preinvex function. therefore, we have

$f(\gamma_{E(x),E(y)}(t))\leq tf(E(x))+(1-t)f(E(y)),$

i.e.,

$f(\gamma_{E(x),E(y)}(t))-f(E(y))\leq t(f(E(x))-f(E(y))).$

On dividing by $t$, we get

$\frac{1}{t}[f(\gamma_{E(x),E(y)}(t))-f(E(y))]\leq(f(E(x))-f(E(y))).$

Now taking limit as $t\rightarrow 0$

$df_{\gamma_{E(x),E(y)}(0)}(\gamma^{\prime}_{E(x),E(y)}(0))\leq
f(E(x))-f(E(y)).$

Therefore,

$df_{E(y)}(\alpha(E(x),E(y))\eta(E(x),E(y)))\leq f(E(x))-f(E(y)).$

Which shows that $f$ is a geodesic $(\alpha,E)$-invex function. ∎

######  Definition 4.1.

(Property (P)) Let $M$ be a Riemannian manifold and
$\gamma_{E(x),E(y)}:[0,1]\rightarrow M$ be a curve on $M$ such that
$\gamma_{E(x),E(y)}(0)=E(y)$ and $\gamma_{E(x),E(y)}(1)=E(x)$. Then
$\gamma_{E(x),E(y)}$ is said to possess the Property (P) with respect to
$x,y\in M$ if

$\gamma^{\prime}_{E(x),E(y)}(s)(t-s)=\alpha(\gamma_{E(x),E(y)}(t),\gamma_{E(x),E(y)}(s))\eta(\gamma_{E(x),E(y)}(t),\gamma_{E(x),E(y)}(s)),\
\forall s,t\in[0,1].$

######  Remark.

If $\alpha=1$, then the above property reduces to the property defined by
Kumari and Jayswal [7]. Agarwal et. al. [1] defined the above property when
$E$ is the identity map. If $\alpha=1$ and $E$ is the identity map, then the
above property is defined by Pini [10].

Let $M$ be a Riemannian manifold and $\gamma_{E(x),E(y)}$ possessing the
Property (P) with respect to $x,y\in M$, then

$\displaystyle\alpha(E(x),E(y))\eta(E(x),E(y))$ $\displaystyle=$
$\displaystyle\alpha(\gamma_{E(x),E(y)}(1),\gamma_{E(x),E(y)}(0))\eta(\gamma_{E(x),E(y)}(1),\gamma_{E(x),E(y)}(0))$
$\displaystyle=$ $\displaystyle\gamma^{\prime}_{E(x),E(y)}(0).$

In this case where $\gamma_{E(x),E(y)}$ is a geodesic, then

$\displaystyle\alpha(\gamma_{E(x),E(y)}(0),\gamma_{E(x),E(y)}(s))\eta(\gamma_{E(x),E(y)}(0),\gamma_{E(x),E(y)}(s))$
$\displaystyle=-s\gamma^{\prime}_{E(x),E(y)}(s)$
$\displaystyle=-sP^{s}_{0,\gamma_{E(x),E(y)}}[\gamma^{\prime}_{E(x),E(y)}(0)]$
$\displaystyle=-sP^{s}_{0,\gamma_{E(x),E(y)}}[\alpha(E(x),E(y))\eta(E(x),E(y))].$

or,

$P^{0}_{s,\gamma_{E(x),E(y)}}[\alpha(E(y),\gamma_{E(x),E(y)}(s))\eta(E(y),\gamma_{E(x),E(y)}(s))]=-s\alpha(E(x),E(y))\eta(E(x),E(y)).$
$\displaystyle\alpha(\gamma_{E(x),E(y)}(1),\gamma_{E(x),E(y)}(s))\eta(\gamma_{E(x),E(y)}(1),\gamma_{E(x),E(y)}(s))$
$\displaystyle=(1-s)\gamma^{\prime}_{E(x),E(y)}(s)$
$\displaystyle=(1-s)P^{s}_{0,\gamma_{E(x),E(y)}}[\gamma^{\prime}_{E(x),E(y)}(0)]$
$\displaystyle=(1-s)P^{s}_{0,\gamma_{E(x),E(y)}}[\alpha(E(x),E(y))\eta(E(x),E(y))].$

or,

$P^{0}_{s,\gamma_{E(x),E(y)}}[\alpha(E(x),\gamma_{E(x),E(y)}(s))\eta(E(x),\gamma_{E(x),E(y)}(s))]=(1-s)\alpha(E(x),E(y))\eta(E(x),E(y)).$

Therefore,

$(C)\begin{cases}P^{0}_{s,\gamma_{E(x),E(y)}}[\alpha(E(y),\gamma_{E(x),E(y)}(s))\eta(E(y),\gamma_{E(x),E(y)}(s))]&=-s\alpha(E(x),E(y))\eta(E(x),E(y)\\\
P^{0}_{s,\gamma_{E(x),E(y)}}[\alpha(E(x),\gamma_{E(x),E(y)}(s))\eta(E(x),\gamma_{E(x),E(y)}(s))]&=(1-s)\alpha(E(x),E(y))\\\
&\eta(E(x),E(y)),\end{cases}$

for all $s\in[0,1]$. The above two conditions we call Condition (C).

######  Theorem 4.3.

Let $S$ be an open subset of $M$ which is geodesic $(\alpha,E)$-invex set with
respect to $\eta:M\times M\rightarrow TM$ and $\alpha:M\times
M\rightarrow\mathbb{R}-\\{0\\}$. Let $f:S\rightarrow\mathbb{R}$ be a
differentiable function and satisfies the condition (C). Then $f$ is a
geodesic $(\alpha,E)$-preinvex on $S$ if $f$ is $(\alpha,E)$-invex on $S$.

###### Proof.

Since $S$ is a geodesic $(\alpha,E)$-invex set with respect to $\eta$ and
$\alpha$, there exists a unique geodesic $\gamma_{E(x),E(y)}:[0,1]\rightarrow
M$ such that

$\gamma_{E(x),E(y)}(0)=E(y),\
\gamma^{\prime}_{E(x),E(y)}(0)=\alpha(E(x),E(y))\eta(E(x),E(y)),\
\gamma_{E(x),E(y)}(t)\in S,\quad\forall t\in[0,1].$

Now, fix $t\in[0,1]$ and set $q=\gamma_{E(x),E(y)}(t)$, then using
$(\alpha,E)$-invexity of $f$ on $S$, we have

(1) $f(E(x))-f(E(q))\geq df_{E(q)}(\alpha(E(x),E(q))\eta(E(x),E(q))).$ (2)
$f(E(y))-f(E(q))\geq df_{E(q)}(\alpha(E(y),E(q))\eta(E(y),E(q))).$

On multiplying (1) by $t$ and (2) by $(1-t)$, respectively, and then adding we
get

$\displaystyle tf(E(x))+(1-t)f(E(y))-f(E(q))$ $\displaystyle\geq$
$\displaystyle df_{E(q)}[t\alpha(E(x),E(q))\eta(E(x),E(q))$
$\displaystyle+(1-t)\alpha(E(y),E(q))\eta(E(y),E(q))].$

By the condition (C), we have

$\displaystyle
t\alpha(E(x),E(q))\eta(E(x),E(q))+(1-t)\alpha(E(y),E(q))\eta(E(y),E(q))$
$\displaystyle=t(1-t)P^{t}_{0,\gamma}[\alpha(E(x),E(y))\eta(E(x),E(y))]-(1-t)tP^{t}_{0,\gamma}[\alpha(E(x),E(y))\eta(E(x),E(y))]$
$\displaystyle=0.$

Now combining the above relation with the inequality (4), we get

$tf(E(x))+(1-t)f(E(y))-f(E(q))\geq 0.$

Which implies that

$f(\gamma_{E(x),E(y)}(t))\leq tf(E(x))+(1-t)f(E(y)).$

Therefore, the function $f$ is a geodesic $(\alpha,E)$-preinvex on $S$. ∎

######  Theorem 4.4.

Let $S\subset M$ be a geodesic $(\alpha,E)$-invex set with respect to
$\eta:M\times M\rightarrow TM$ and $\alpha:M\times M\rightarrow R$. A function
$f:S\rightarrow R$ is a geodesic $(\alpha,E)$-preinvex and $\phi:I\rightarrow
R$ is an increasing geodesic pre-invex such that range $f\subset I$, where $I$
is any interval. Then the composite function $\phi\circ f$ is a geodesic
$(\alpha,E)$-preinvex on S.

###### Proof.

Since $S$ is a geodesic $(\alpha,E)$-invex set with respect to $\eta$ and
$\alpha$, there exists a unique geodesic $\gamma_{E(x),E(y)}:[0,1]\rightarrow
M$ such that

$\gamma_{E(x),E(y)}(0)=E(y),\
\gamma^{\prime}_{E(x),E(y)}(0)=\alpha(E(x),E(y))\eta(E(x),E(y)),\
\gamma_{E(x),E(y)}(t)\in S,\quad\forall t\in[0,1].$

By the defnition of geodesic $(\alpha,E)$-preinvex function, we have

$f(\gamma_{E(x),E(y)}(t))\leq tf(E(x))+(1-t)f(E(y)).$

As $\phi$ is an increasing geodesic pre-invex function, we get

$\phi\circ f(\gamma_{E(x),E(y)}(t))\leq\phi(tf(E(x))+(1-t)f(E(y)))$
$\displaystyle(\phi\circ f)(\gamma_{E(x),E(y)}(t))$ $\displaystyle\leq$
$\displaystyle t\phi(f(E(x)))+(1-t)\phi(f(E(y)))$ $\displaystyle=$
$\displaystyle t(\phi\circ f)(E(x))+(1-t)(\phi\circ f)(E(y)).$

Therefore, $\phi\circ f$ is a geodesic $(\alpha,E)$-preinvex function on S. ∎

######  Theorem 4.5.

Let $S\subset M$ be a geodesic $(\alpha,E)$-invex set with respect to
$\eta:M\times M\rightarrow TM$ and $\alpha:M\times M\rightarrow R$. If
$f_{i}:S\rightarrow R$, $j\in J$ are geodesic $(\alpha,E)$-preinvex functions
on S such that $\sup_{j\in J}f_{j}$ exist in $\mathbb{R}$, then the function
$f:S\rightarrow R$ defined by

$f(x)=\sup_{j\in J}f_{j}(x),\text{ for }x\in S,$

is a geodesic $(\alpha,E)$-preinvex function on S.

###### Proof.

Since $S$ is a geodesic $(\alpha,E)$-invex set with respect to $\eta$ and
$\alpha$, there exists a unique geodesic $\gamma_{E(x),E(y)}:[0,1]\rightarrow
M$ such that

$\gamma_{E(x),E(y)}(0)=E(y),\
\gamma^{\prime}_{E(x),E(y)}(0)=\alpha(E(x),E(y))\eta(E(x),E(y)),\
\gamma_{E(x),E(y)}(t)\in S,\quad\forall t\in[0,1].$

From the geodesic $(\alpha,E)$-preinvexity of $f_{j}$, $j\in J$, we have

$f_{j}(\gamma_{E(x),E(y)}(t))\leq tf_{j}(E(x))+(1-t)f_{j}(E(y)).$

Then,

$\displaystyle\sup_{j\in J}f_{j}(\gamma_{E(x),E(y)}(t))$ $\displaystyle\leq$
$\displaystyle\sup_{j\in J}\Big{(}tf_{j}(E(x))+(1-t)f_{j}(E(y))\Big{)}$
$\displaystyle\leq$ $\displaystyle t\Big{(}\sup_{j\in
J}f_{j}(E(x))\Big{)}+(1-t)\Big{(}\sup_{j\in J}f_{j}(E(y))\Big{)}$
$\displaystyle=$ $\displaystyle tf(E(x))+(1-t)f(E(y)).$

Therefore, we get

$f(\gamma_{E(x),E(y)}(t))\leq tf(E(x))+(1-t)f(E(y)).$

This proves the $(\alpha,E)$-preinvexity of $f$. ∎

######  Theorem 4.6.

Let $S$ be a geodesic $(\alpha,E)$-invex set with respect to $\eta:M\times
M\rightarrow TM$ and $\alpha:M\times M\rightarrow\mathbb{R}$. Suppose
$F:S\times S\rightarrow\mathbb{R}$ is a continuous geodesic
$(\alpha,E)$-preinvex, i.e., $F$ is geodesic $(\alpha,E)$-preinvex with
respect to each variable. Then the function $f:S\rightarrow\mathbb{R}$ defined
by

$f(p)=\inf_{q\in S}F(p,q),$

is a geodesic $(\alpha,E)$-preinvex function on $K$.

###### Proof.

Suppose $\epsilon>0$ is an arbitrary small number and $p_{0},p_{1}\in S$.
Since $S$ is a geodesic $(\alpha,E)$-invex set with respect to $\eta$ and
$\alpha$, there exists a unique geodesic
$\gamma_{E(p_{0}),E(p_{1})}:[0,1]\rightarrow M$ such that, for all
$t\in[0,1],$

$\gamma_{E(p_{0}),E(p_{1})}(0)=E(p_{1}),\
\gamma^{\prime}_{E(p_{0}),E(p_{1})}(0)=\alpha(E(p_{0}),E(p_{1}))\eta(E(p_{0}),E(p_{1})),\
\gamma_{E(p_{0}),E(p_{1})}(t)\in S.$

Now from the definition of $f$, we get, there exists $q_{0},q_{1}\in S$ such
that

$F(p_{1},q_{1})<f(p_{1})+\epsilon,\ F(p_{0},q_{0})<f(p_{0})+\epsilon.$

By the geodesic $(\alpha,E)$-set with respect to $\eta$ and $\alpha$, there
exists exactly one geodesic $\lambda_{E(q_{0}),E(q_{1})}:[0,1]\rightarrow M$
such that, for all $s\in[0,1],$

$\lambda_{E(q_{0}),E(q_{1})}(0)=E(q_{1}),\
\lambda^{\prime}_{E(q_{0}),E(q_{1})}(0)=\alpha(E(q_{0}),E(q_{1}))\eta(E(q_{0}),E(q_{1})),\
\lambda_{E(q_{0}),E(q_{1})}(s)\in S.$

Hence, the curve
$\Gamma=(\gamma_{E(p_{0}),E(p_{1})},\lambda_{E(q_{0}),E(q_{1})}):[0,1]\rightarrow
M\times M$ is a geodesic in $S\times S$, with

$\displaystyle\Gamma(0)=(E(p_{1}),E(q_{1})\text{ and }$
$\displaystyle\Gamma^{\prime}(0)=(\alpha(E(p_{0}),E(p_{1}))\eta(E(p_{0}),E(p_{1})),\alpha(E(q_{0}),E(q_{1}))\eta(E(q_{0}),E(q_{1}))).$

Since $\Gamma$ is a geodesic in $S\times S$,

$\Gamma(s)=(\gamma_{E(p_{0}),E(p_{1})}(s),\lambda_{E(q_{0}),E(q_{1})}(s))\in
S\times S\ \forall s\in[0,1].$

Now from the definition of $f$ and the geodesic $(\alpha,E)$-preinvexity of
$F$, we get

$\displaystyle f(\gamma_{E(p_{0}),E(p_{1})}(s))$ $\displaystyle=$
$\displaystyle\inf_{q\in S}F(\gamma_{E(p_{0}),E(p_{1})}(s),E(q))$
$\displaystyle\leq$ $\displaystyle
F(\gamma_{E(p_{0}),E(p_{1})}(s),\lambda_{E(q_{0}),E(q_{1})}(s))$
$\displaystyle\leq$ $\displaystyle
sF(E(p_{0}),E(q_{0}))+(1-s)F(E(p_{1}),E(q_{1}))$ $\displaystyle\leq$
$\displaystyle s(f(E(p_{0}))+\epsilon)+(1-s)(f(E(p_{1}))+\epsilon)$
$\displaystyle=$ $\displaystyle sf(E(p_{0}))+(1-s)f(E(p_{1}))+\epsilon.$

Since $\epsilon$ is arbitrary number, therefore

$f(\gamma_{E(p_{0}),E(p_{1})}(s))\leq sf(E(p_{0}))+(1-s)f(E(p_{1})).$

Hence, we get our theorem. ∎

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