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- W2024952253 abstract "Let <svg style=vertical-align:-0.1092pt;width:12.725px; id=M1 height=11.3125 version=1.1 viewBox=0 0 12.725 11.3125 width=12.725 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,11.113)><path id=x211D d=M751 18q-1 -4 -2 -13t-2 -13q-24 0 -35 1q-57 3 -92.5 23t-69.5 68q-21 30 -101 160q-13 22 -30.5 31t-54.5 9h-32v-159q0 -62 14 -77t74 -20v-28h-382v28q62 5 76 19.5t14 77.5v400q0 63 -14 77.5t-76 19.5v28h372q109 0 159 -34q67 -42 67 -131q0 -115 -127 -172
q27 -52 89 -144q49 -71 81 -104q31 -36 72 -47zM543 469q0 76 -42 112t-104 36q-41 0 -54 -10q-11 -7 -11 -44v-247h48q73 0 108 29q55 42 55 124zM290 131v390q0 62 -7.5 79.5t-30.5 17.5q-24 0 -31.5 -17.5t-7.5 -79.5v-390q0 -62 7.5 -79.5t31.5 -17.5q23 0 30.5 17.5
t7.5 79.5z /></g> </svg> be the set of real numbers, <svg style=vertical-align:-2.30685pt;width:143.0625px; id=M2 height=16.987499 version=1.1 viewBox=0 0 143.0625 16.987499 width=143.0625 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,14.05)><use xlink:href=#x211D/></g> <g transform=matrix(.012,-0,0,-.012,12.663,5.887)><path id=x2B d=M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z /></g> <g transform=matrix(.017,-0,0,-.017,24.988,14.05)><path id=x3D d=M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z /></g><g transform=matrix(.017,-0,0,-.017,39.674,14.05)><path id=x7B d=M293 -169q-145 0 -145 128q0 35 4 106t3 104q0 17 -2 28t-10 25.5t-28 23t-51 10.5v29q31 2 51 10.5t28 23t10 25.5t2 28q0 31 -4 101t-3 104q0 132 145 132v-28q-51 -7 -67.5 -30t-16.5 -63q0 -32 6.5 -99.5t6.5 -100.5q0 -97 -83 -115v-4q83 -20 83 -117
q0 -34 -6.5 -99.5t-6.5 -96.5q0 -41 16.5 -65.5t67.5 -31.5v-28z /></g><g transform=matrix(.017,-0,0,-.017,45.573,14.05)><path id=x1D465 d=M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37
q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z /></g><g transform=matrix(.017,-0,0,-.017,59.801,14.05)><path id=x2208 d=M448 1h-83q-118 0 -201.5 74.5t-83.5 179.5t83.5 179.5t201.5 74.5h83v-50h-84q-87 0 -150.5 -51.5t-73.5 -127.5h308v-50h-308q10 -76 73.5 -127.5t150.5 -51.5h84v-50z /></g><g transform=matrix(.017,-0,0,-.017,73.502,14.05)><use xlink:href=#x211D/></g><g transform=matrix(.017,-0,0,-.017,90.824,14.05)><path id=x2223 d=M162 -163h-61v866h61v-866z /></g><g transform=matrix(.017,-0,0,-.017,100.003,14.05)><use xlink:href=#x1D465/></g><g transform=matrix(.017,-0,0,-.017,114.231,14.05)><path id=x3E d=M512 230l-437 -233v58l378 199v2l-378 200v58l437 -233v-51z /></g><g transform=matrix(.017,-0,0,-.017,128.935,14.05)><path id=x30 d=M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z /></g><g transform=matrix(.017,-0,0,-.017,137.094,14.05)><path id=x7D d=M283 255q-31 -2 -51 -10.5t-28 -23.5t-10 -27t-2 -30q0 -28 4 -97.5t4 -107.5q0 -128 -146 -128v28q50 7 67.5 31t17.5 63q0 33 -7 100t-7 98q0 96 84 117v4q-84 20 -84 115q0 34 7 102.5t7 98.5q0 40 -17.5 63t-67.5 30v28q28 0 50 -4t45.5 -16.5t37 -40t13.5 -68.5
q0 -37 -4 -106.5t-4 -98.5q0 -18 2 -30t10 -27t28 -23.5t51 -10.5v-29z /></g> </svg>, <svg style=vertical-align:-3.30472pt;width:45.950001px; id=M3 height=15.2875 version=1.1 viewBox=0 0 45.950001 15.2875 width=45.950001 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,11.113)><path id=x1D716 d=M401 397q0 -24 -24 -40q-18 -13 -31 3q-36 45 -93 45q-48 0 -86.5 -43t-51.5 -112l206 6l-8 -48l-203 6v-25q0 -68 26.5 -107t73.5 -39q73 0 148 64l16 -22q-40 -44 -91 -70.5t-102 -26.5q-75 0 -116.5 45t-41.5 136q0 125 99 208q36 30 87 50.5t93 20.5q46 0 72.5 -16
t26.5 -35z /></g><g transform=matrix(.017,-0,0,-.017,11.979,11.113)><use xlink:href=#x2208/></g><g transform=matrix(.017,-0,0,-.017,25.68,11.113)><use xlink:href=#x211D/></g> <g transform=matrix(.012,-0,0,-.012,38.288,15.188)><use xlink:href=#x2B/></g> </svg>, and <svg style=vertical-align:-3.56265pt;width:122.075px; id=M4 height=18.549999 version=1.1 viewBox=0 0 122.075 18.549999 width=122.075 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,14.05)><path id=x1D453 d=M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19
q34 34 71 253l54 315h-90l-3 12l31 30h70q28 138 90 204q35 37 75 57.5t70 20.5q26 0 45 -14t19 -28z /></g><g transform=matrix(.017,-0,0,-.017,10.976,14.05)><path id=x2C d=M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z /></g><g transform=matrix(.017,-0,0,-.017,17.673,14.05)><path id=x1D454 d=M546 430q-14 -33 -37.5 -140t-38.5 -209q-29 -195 -156 -289q-29 -21 -70 -37t-78 -16q-62 0 -102.5 37t-40.5 69q0 27 26 48q19 15 25 -3q13 -43 46.5 -69t87.5 -26q55 0 83 21q29 22 50 65t43 140q14 64 29 163h-2q-55 -81 -149 -152q-59 -44 -102 -44q-23 0 -43.5 30
t-20.5 85q0 74 32.5 145.5t84.5 117.5q40 35 100 58.5t117 23.5q21 0 65 -8q30 -6 44 -6zM456 386q-40 16 -81 16q-29 0 -64 -13q-42 -17 -85 -97q-19 -36 -31.5 -86t-12.5 -86q0 -64 30 -64q20 0 52.5 21t60.5 52q74 81 95 123q26 62 36 134z /></g><g transform=matrix(.017,-0,0,-.017,27.345,14.05)><use xlink:href=#x2C/></g><g transform=matrix(.017,-0,0,-.017,34.06,14.05)><path id=x210E d=M499 88q-41 -46 -83.5 -73t-68.5 -27q-41 0 -19 98l52 225q10 43 8 59.5t-16 16.5q-37 0 -112 -65t-117 -135q-14 -42 -38 -179q-36 -7 -75 -20l-7 8l127 594q8 40 2.5 49t-35.5 9h-35l5 25q36 3 71.5 13t56.5 18t25 8q11 0 3 -38l-88 -417h2q77 102 195 167q44 24 79 24
q61 0 30 -130l-52 -220q-8 -33 4 -33q8 0 31 14.5t43 33.5z /></g><g transform=matrix(.017,-0,0,-.017,47.557,14.05)><path id=x2236 d=M223 355q0 -25 -15.5 -41.5t-39.5 -16.5t-40 16.5t-16 41.5t16 41t40 16t39.5 -16t15.5 -41zM223 46q0 -26 -16 -42t-40 -16t-39.5 16t-15.5 42q0 24 16 40.5t39 16.5q24 0 40 -16.5t16 -40.5z /></g><g transform=matrix(.017,-0,0,-.017,57.943,14.05)><use xlink:href=#x211D/></g> <g transform=matrix(.012,-0,0,-.012,70.55,5.887)><use xlink:href=#x2B/></g> <g transform=matrix(.017,-0,0,-.017,85.712,14.05)><path id=x2192 d=M901 255q-71 -62 -185 -187l-22 15l102 147h-727v50h727l-102 147l22 15q114 -125 185 -187z /></g><g transform=matrix(.017,-0,0,-.017,109.477,14.05)><path id=x2102 d=M708 165q-34 -105 -63 -154q-134 -26 -204 -26q-99 0 -177.5 29t-125 77t-70.5 105.5t-24 119.5q0 157 120 253t307 96q40 0 79.5 -5t76.5 -13.5t40 -8.5l23 -151l-33 -7q-23 80 -72 114.5t-142 34.5q-79 0 -123.5 -86t-44.5 -207q0 -140 49 -227.5t125 -87.5
q84 0 133 36.5t94 117.5q6 -2 17.5 -5.5t14.5 -4.5zM313 50q-81 98 -81 286q0 175 79 267q-79 -35 -122 -107.5t-43 -159.5q0 -99 44.5 -174t122.5 -112z /></g> </svg>. As classical and <svg style=vertical-align:-0.0pt;width:22.5875px; id=M5 height=13.6125 version=1.1 viewBox=0 0 22.5875 13.6125 width=22.5875 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,13.55)><path id=x1D43F d=M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z /></g> <g transform=matrix(.012,-0,0,-.012,9.763,5.388)><path id=x221E d=M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196
q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z /></g> </svg> versions of the Hyers-Ulam stability of the logarithmic type functional equation in a restricted domain, we consider the following inequalities: <svg style=vertical-align:-3.56265pt;width:282.36249px; id=M6 height=16.6625 version=1.1 viewBox=0 0 282.36249 16.6625 width=282.36249 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,12.162)><path id=x7C d=M162 -163h-61v866h61v-866z /></g><g transform=matrix(.017,-0,0,-.017,4.533,12.162)><use xlink:href=#x1D453/></g><g transform=matrix(.017,-0,0,-.017,15.446,12.162)><path id=x28 d=M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z /></g><g transform=matrix(.017,-0,0,-.017,21.328,12.162)><use xlink:href=#x1D465/></g><g transform=matrix(.017,-0,0,-.017,34.604,12.162)><use xlink:href=#x2B/></g><g transform=matrix(.017,-0,0,-.017,48.356,12.162)><path id=x1D466 d=M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26
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