Simpson’s Integral Inequalities for Twice Differentiable Convex Functions

<div>Plot illustrations for inequality (<a href="https://static.hindawi.com/articles/mpe/volume-2020/1936461/figures/#EEq44">44</a>). (a) For <svg height="6.1673pt" id="M350" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 9.39034 6.1673" width="9.39034pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M689 332C689 394 670 448 646 448C620 448 597 421 597 396C597 386 600 381 608 372C619 359 620 334 620 315C620 150 538 45 454 45C414 45 386 67 386 122C386 138 388 158 394 180L457 426L452 432L377 416L315 156C302 100 259 45 216 45C176 45 148 67 148 122C148 133 152 158 156 180C162 212 173 259 194 332C201 357 206 384 206 405C206 430 198 448 174 448C125 448 66 406 23 342L43 319C84 368 110 383 121 383C126 383 128 382 128 377C128 370 127 359 122 343C99 268 84 204 77 156C74 137 70 111 70 104C70 25 125 -12 180 -12C228 -12 276 12 319 50C338 8 378 -12 418 -12C549 -12 689 166 689 332Z"></path></g></svg> (<i>x</i>; <i>y</i>) and <i>W</i> (<i>x</i>; <i>y</i>). (b) For <span class="nowrap"><svg height="12.7178pt" id="M351" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 93.6373 12.7178" width="93.6373pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M962 650H739L732 622L760 619C812 613 819 604 798 552C760 457 671 267 606 131H604L511 638H480L237 131H233L183 554C177 607 183 614 226 619L248 622L257 650H24L17 622C88 615 92 611 103 524L173 -11H203L450 491H453L543 -11H575L839 529C882 609 886 613 953 622L962 650Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.669,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,17.167,0)"><path d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.434,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,29.578,0)"><path d="M556 393C556 426 537 448 514 448C496 448 478 435 471 420C468 414 462 401 466 394C473 382 480 368 480 346C480 268 392 143 338 67H336C329 163 319 253 309 330S286 448 254 448C215 448 160 383 127 337L143 311C167 344 200 373 208 373S222 365 229 320C246 214 264 66 268 -20C219 -83 131 -171 17 -239L25 -261L137 -235C248 -110 273 -79 335 8C384 77 481 215 520 287C544 332 556 363 556 393Z"></path></g><g transform="matrix(.013,0,0,-0.013,37.107,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g><g transform="matrix(.013,0,0,-0.013,44.51,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,55.047,0)"><path d="M689 332C689 394 670 448 646 448C620 448 597 421 597 396C597 386 600 381 608 372C619 359 620 334 620 315C620 150 538 45 454 45C414 45 386 67 386 122C386 138 388 158 394 180L457 426L452 432L377 416L315 156C302 100 259 45 216 45C176 45 148 67 148 122C148 133 152 158 156 180C162 212 173 259 194 332C201 357 206 384 206 405C206 430 198 448 174 448C125 448 66 406 23 342L43 319C84 368 110 383 121 383C126 383 128 382 128 377C128 370 127 359 122 343C99 268 84 204 77 156C74 137 70 111 70 104C70 25 125 -12 180 -12C228 -12 276 12 319 50C338 8 378 -12 418 -12C549 -12 689 166 689 332Z"></path></g><g transform="matrix(.013,0,0,-0.013,64.304,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,68.802,0)"><path d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g><g transform="matrix(.013,0,0,-0.013,76.069,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,81.212,0)"><path d="M556 393C556 426 537 448 514 448C496 448 478 435 471 420C468 414 462 401 466 394C473 382 480 368 480 346C480 268 392 143 338 67H336C329 163 319 253 309 330S286 448 254 448C215 448 160 383 127 337L143 311C167 344 200 373 208 373S222 365 229 320C246 214 264 66 268 -20C219 -83 131 -171 17 -239L25 -261L137 -235C248 -110 273 -79 335 8C384 77 481 215 520 287C544 332 556 363 556 393Z"></path></g><g transform="matrix(.013,0,0,-0.013,88.741,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg>.</span></div>

Mathematical Problems in Engineering

fig2

Figure 2

Figure 2: Simpson’s Integral Inequalities for Twice Differentiable Convex Functions 