Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M1" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 9.97754 8.68572" width="9.97754pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-153" d="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z"/></g></svg> be a complex domain and let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M2" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 8.02022 8.68572" width="8.02022pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-71" d="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z"/></g></svg> be a reflexive BK space with AK such that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.2681pt" id="M3" height="14.1923pt" version="1.1" viewBox="-0.0498162 -11.9242 52.8121 14.1923" width="52.8121pt"><g transform="matrix(.013,0,0,-0.013,2.003,-2.897)"><path id="g119-17" d="M470 557C393 591 314 624 246 666H224C156 624 76 591 0 557L12 532C86 550 163 573 235 596C309 572 384 550 458 532L470 557Z"/></g><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-71"/></g><g transform="matrix(.013,0,0,-0.013,11.552,0)"><path id="g117-77" d="M512 1V51H381C244 51 133 142 133 255S244 459 381 459H512V509H382C213 509 75 395 75 255S213 1 382 1H512Z"/></g><g transform="matrix(.013,0,0,-0.013,22.815,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g><g transform="matrix(.013,0,0,-0.013,33.738,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.013,0,0,-0.013,38.236,0)"><use xlink:href="#g113-153"/></g><g transform="matrix(.013,0,0,-0.013,48.078,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg> and the functional of evaluation at <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M4" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 7.30254 9.49473" width="7.30254pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-233" d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"/></g></svg> is bounded for all <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.6370001pt" id="M5" height="9.92533pt" version="1.1" viewBox="-0.0498162 -9.28833 31.3339 9.92533" width="31.3339pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-233"/></g><g transform="matrix(.013,0,0,-0.013,10.806,0)"><path id="g117-173" d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z"/></g><g transform="matrix(.013,0,0,-0.013,21.302,0)"><use xlink:href="#g113-153"/></g></svg>. We will investigate the cyclicity for the adjoint of a weighted composition operator acting on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04979992pt" id="M6" height="11.974pt" version="1.1" viewBox="-0.0498162 -11.9242 8.02022 11.974" width="8.02022pt"><g transform="matrix(.013,0,0,-0.013,2.003,-2.897)"><use xlink:href="#g119-17"/></g><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-71"/></g></svg>.

Leila Bagheri

Bahmann Yousefi

Cyclicity of Special Operators on a BK with AK Space

This paper is an expository devoted to an important class of real-valued functions introduced by L&#xF6;wner, namely, operator monotone functions. This concept is closely related to operator convex/concave functions. Various characterizations for such functions are given from the viewpoint of differential analysis in terms of matrix of divided differences. From the viewpoint of operator inequalities, various characterizations and the relationship between operator monotonicity and operator convexity are given by Hansen and Pedersen. In the viewpoint of measure theory, operator monotone functions on the nonnegative reals admit meaningful integral representations with respect to Borel measures on the unit interval. Furthermore, Kubo-Ando theory asserts the correspondence between operator monotone functions and operator means.

Pattrawut Chansangiam

A Survey on Operator Monotonicity, Operator Convexity, and Operator Means

The authors introduce the concept of harmonically <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M2" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 29.4723 11.5564" width="29.4723pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path id="g113-116" d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z"/></g><g transform="matrix(.013,0,0,-0.013,9.373,0)"><path id="g113-45" 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"/></g><g transform="matrix(.013,0,0,-0.013,14.516,0)"><path id="g113-110" d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"/></g><g transform="matrix(.013,0,0,-0.013,24.774,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-convex functions in second sense and establish some Ostrowski type inequalities of these classes of functions.

Imran Abbas Baloch

&#x130;mdat &#x130;&#x15f;can

Some Ostrowski Type Inequalities for Harmonically <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 29.4723 11.5564" width="29.4723pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" /></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path id="g113-116" d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z" /></g><g transform="matrix(.013,0,0,-0.013,9.373,0)"><path id="g113-45" 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" /></g><g transform="matrix(.013,0,0,-0.013,14.516,0)"><path id="g113-110" d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z" /></g><g transform="matrix(.013,0,0,-0.013,24.774,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" /></g></svg>-Convex Functions in Second Sense

Integral equalities involving integrals of the logarithm of the Riemann <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.15065pt" id="M2" height="8.11156pt" version="1.1" viewBox="-0.0498162 -5.96091 5.41045 8.11156" width="5.41045pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-252" d="M384 393C384 425 346 448 294 448C244 448 200 431 150 399C75 351 23 271 23 176C23 90 52 45 132 19C150 13 166 9 195 2C245 -10 260 -27 260 -55C260 -92 234 -127 218 -145L237 -159C271 -137 333 -93 333 -18C333 43 283 62 228 74C199 80 186 84 172 89C116 109 101 150 101 210C101 318 164 400 241 400C276 400 301 390 327 356C334 347 338 343 346 345C367 349 384 374 384 393Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="407" vert-adv-y="407"/></g></svg>-function with exponential weight functions are introduced, and it is shown that an infinite number of them are equivalent to the Riemann hypothesis. Some of these equalities are tested numerically. The possible contribution of the Riemann function zeroes nonlying on the critical line is rigorously estimated and shown to be extremely small, in particular, smaller than nine milliards of decimals for the maximal possible weight function exp(<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3499298pt" id="M3" height="8.69875pt" version="1.1" viewBox="-0.0498162 -8.34882 26.445 8.69875" width="26.445pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"/></g><g transform="matrix(.013,0,0,-0.013,7.631,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"/></g><g transform="matrix(.013,0,0,-0.013,13.871,0)"><path id="g113-238" d="M574 449L545 460C526 432 516 430 487 430C404 430 311 435 226 435C104 435 56 379 25 341L43 318C81 354 121 372 181 372C161 246 87 53 23 3L30 -12C48 -12 88 -4 113 11C157 75 207 248 232 371L386 367L326 109C321 86 318 66 318 50C318 4 339 -12 366 -12C410 -12 461 21 505 69L492 96C467 79 434 60 418 60C406 60 400 78 411 147C422 217 439 300 457 366C487 366 524 367 536 370C547 385 558 408 574 449Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="591" vert-adv-y="591"/></g><g transform="matrix(.013,0,0,-0.013,21.816,0)"><path id="g113-117" d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="341" vert-adv-y="341"/></g></svg>). We also show how certain Fourier transforms of the logarithm of the Riemann zeta-function taken along the real (demi)axis are expressible via elementary functions plus logarithm of the gamma-function and definite integrals thereof, as well as certain sums over trivial and nontrivial Riemann function zeroes.

Sergey K. Sekatskii

Stefano Beltraminelli

Danilo Merlini

On Equalities Involving Integrals of the Logarithm of the Riemann <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-2.15065pt" height="8.11156pt" version="1.1" viewBox="-0.0498162 -5.96091 5.41045 8.11156" width="5.41045pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-252" d="M384 393C384 425 346 448 294 448C244 448 200 431 150 399C75 351 23 271 23 176C23 90 52 45 132 19C150 13 166 9 195 2C245 -10 260 -27 260 -55C260 -92 234 -127 218 -145L237 -159C271 -137 333 -93 333 -18C333 43 283 62 228 74C199 80 186 84 172 89C116 109 101 150 101 210C101 318 164 400 241 400C276 400 301 390 327 356C334 347 338 343 346 345C367 349 384 374 384 393Z" /></g></svg>-Function with Exponential Weight Which Are Equivalent to the Riemann Hypothesis

We discuss properties of modified Baskakov-Durrmeyer-Stancu (BDS) operators with parameter <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42938pt" id="M1" height="11.7782pt" version="1.1" viewBox="-0.0498162 -8.34882 27.8238 11.7782" width="27.8238pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-225" d="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 478 372Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="501" vert-adv-y="501"/></g><g transform="matrix(.013,0,0,-0.013,10.148,0)"><path id="g117-92" d="M512 230V281L75 514V456L453 256V254L75 55V-3L512 230Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"/></g><g transform="matrix(.013,0,0,-0.013,21.411,0)"><path id="g113-49" d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"/></g></svg>. We compute the moments of these modified operators. Also, we establish pointwise convergence, Voronovskaja type asymptotic formula, and an error estimation in terms of second order modification of continuity of the function for the operators <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-6.1814pt" id="M2" height="18.4396pt" version="1.1" viewBox="-0.0498162 -12.2582 49.5722 18.4396" width="49.5722pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-67" d="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="601" vert-adv-y="601"/></g><g transform="matrix(.0091,0,0,-0.0091,7.813,-5.741)"><path id="g50-223" d="M558 106L533 129C501 84 476 67 464 67C446 67 435 116 414 239C457 297 508 367 550 440L539 451L484 438C458 390 429 338 404 300H402C375 407 351 451 285 451C168 451 24 312 24 153C24 56 66 -12 132 -12C206 -12 284 64 345 156H347C367 24 386 -12 422 -12C457 -12 503 7 558 106ZM338 206C272 99 215 56 173 56C140 56 116 96 116 170C116 303 194 405 255 405C304 405 324 299 338 206Z"/><glyph.data ascent="3443" descent="-2856" horiz-adv-x="582" vert-adv-y="582"/></g><g transform="matrix(.0091,0,0,-0.0091,13.11,-5.741)"><path id="g50-45" d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"/><glyph.data ascent="3443" descent="-2856" horiz-adv-x="237" vert-adv-y="237"/></g><g transform="matrix(.0091,0,0,-0.0091,15.267,-5.741)"><path id="g50-224" d="M563 583C563 664 501 710 433 710C381 710 334 692 285 648C212 582 178 506 149 350L74 -56C49 -190 36 -222 24 -226L30 -257C52 -256 95 -247 121 -229C131 -222 134 -206 165 -21L219 303C254 510 296 665 397 665C447 665 481 626 481 572C481 512 446 457 390 420C367 405 351 400 323 397L302 349C398 338 466 285 466 197C466 117 419 45 306 45C268 45 224 59 198 74L186 50C197 20 228 -14 274 -14C318 -14 359 3 401 26C478 68 554 144 554 225C554 322 493 367 412 396C490 446 563 500 563 583Z"/><glyph.data ascent="3443" descent="-2856" horiz-adv-x="587" vert-adv-y="587"/></g><g transform="matrix(.0091,0,0,-0.0091,7.748,3.784)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/><glyph.data ascent="3443" descent="-2856" horiz-adv-x="512" vert-adv-y="512"/></g><g transform="matrix(.0091,0,0,-0.0091,12.407,3.784)"><path id="g50-45" d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"/><glyph.data ascent="3443" descent="-2856" horiz-adv-x="237" vert-adv-y="237"/></g><g transform="matrix(.0091,0,0,-0.0091,14.564,3.784)"><path id="g50-225" d="M493 373C493 420 472 451 445 451C421 451 400 435 400 410C400 404 402 400 406 394S417 371 417 350C417 259 317 128 256 55H254C260 126 256 262 238 340C219 424 194 451 163 451C128 451 78 413 24 329L52 302C89 353 106 370 121 370C129 370 137 363 147 336C186 231 196 73 184 -16C152 -88 120 -192 112 -238L130 -257C157 -253 213 -227 231 -213C231 -187 234 -77 240 -23C251 -6 277 30 309 69C424 208 493 299 493 373Z"/><glyph.data ascent="3443" descent="-2856" horiz-adv-x="517" vert-adv-y="517"/></g><g transform="matrix(.013,0,0,-0.013,21.154,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"/></g><g transform="matrix(.013,0,0,-0.013,25.652,0)"><path id="g113-103" d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="642" vert-adv-y="642"/></g><g transform="matrix(.013,0,0,-0.013,32.438,0)"><path id="g113-45" 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"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"/></g><g transform="matrix(.013,0,0,-0.013,37.581,0)"><path id="g113-121" 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"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="559" vert-adv-y="559"/></g><g transform="matrix(.013,0,0,-0.013,44.848,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"/></g></svg>.

Prashantkumar G. Patel

Vishnu Narayan Mishra

On Simultaneous Approximation of Modified Baskakov-Durrmeyer Operators

Manpreet Kaur

Sushma Gupta

Sukhjit Singh

Geometric Properties of a Class of Analytic Functions Defined by a Differential Inequality

The aim of this paper is to investigate an integral mean value theorem proposed by one of the references of this paper. Unfortunately, the proof contains a gap. First, we present a counterexample which shows that this theorem fails in this form. Then, we present two improved versions of this theorem. The stability of the mean point arising from the second result concludes this paper.

Monea Mihai

An Integral Mean Value Theorem concerning Two Continuous Functions and Its Stability

Various investigators have studied the degree of approximation of a function using different summability (Ces&#xe1;ro means of order <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M1" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 7.51131 6.1673" width="7.51131pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g></svg>: <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.291101pt" id="M2" height="11.927pt" version="1.1" viewBox="-0.0498162 -8.6359 14.5651 11.927" width="14.5651pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-68" d="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z"/></g><g transform="matrix(.0091,0,0,-0.0091,8.619,3.132)"><path id="g50-223" d="M558 106L533 129C501 84 476 67 464 67C446 67 435 116 414 239C457 297 508 367 550 440L539 451L484 438C458 390 429 338 404 300H402C375 407 351 451 285 451C168 451 24 312 24 153C24 56 66 -12 132 -12C206 -12 284 64 345 156H347C367 24 386 -12 422 -12C457 -12 503 7 558 106ZM338 206C272 99 215 56 173 56C140 56 116 96 116 170C116 303 194 405 255 405C304 405 324 299 338 206Z"/></g></svg>, Euler <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.4742pt" id="M3" height="14.1101pt" version="1.1" viewBox="-0.0498162 -8.6359 13.1285 14.1101" width="13.1285pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-70" d="M593 650H142L136 622C219 616 224 611 209 531L130 120C114 39 108 34 23 28L17 0H501C518 32 557 130 568 158L540 169C519 132 495 93 469 70C439 43 409 35 332 35C272 35 241 37 225 50C211 61 211 89 222 149L253 320H350C441 320 443 312 441 239H471L510 434H481C455 366 448 358 360 358H260L303 572C311 613 312 615 348 615H431C509 615 522 603 532 581C544 555 546 531 545 491L574 493C583 555 592 631 593 650Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.943,3.132)"><path id="g50-114" d="M481 430L460 436C440 442 393 451 366 451C348 451 325 448 313 446C267 437 195 409 147 367C79 308 24 211 24 105C24 37 56 -12 93 -12C121 -12 156 1 199 32C246 66 313 122 349 168H351L286 -147C274 -205 261 -215 210 -224L199 -226L193 -255L438 -243L442 -214L417 -212C364 -208 359 -200 370 -140L433 204C455 324 469 389 481 430ZM391 385C382 336 366 260 358 238C321 178 209 59 143 59C129 59 115 83 115 131C115 207 152 317 220 373C243 392 282 403 314 403C348 403 374 395 391 385Z"/></g></svg>, and N&#xf6;rlund <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.4742pt" id="M4" height="14.1101pt" version="1.1" viewBox="-0.0498162 -8.6359 16.1505 14.1101" width="16.1505pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-79" d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z"/></g><g transform="matrix(.0091,0,0,-0.0091,10.062,3.132)"><path id="g50-113" d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z"/></g></svg>) means of its Fourier-Laguerre series at the point <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M5" height="8.55521pt" version="1.1" viewBox="-0.0498162 -8.34882 28.5806 8.55521" width="28.5806pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-121" 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"/></g><g transform="matrix(.013,0,0,-0.013,10.899,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.013,0,0,-0.013,22.162,0)"><path id="g113-49" d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"/></g></svg> after replacing the continuity condition in Szeg&#xf6; theorem by much lighter conditions. The product summability methods are more powerful than the individual summability methods and thus give an approximation for wider class of functions than the individual methods. This has motivated us to investigate the error estimation of a function by <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.47417pt" id="M6" height="14.7625pt" version="1.1" viewBox="-0.0498162 -9.28833 39.2563 14.7625" width="39.2563pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path id="g113-85" d="M620 675H597C578 656 570 650 541 650H144C112 650 104 653 94 675H72C59 618 42 552 23 493L53 491C71 534 88 564 105 585C124 608 144 615 238 615H290L197 121C182 40 174 34 88 28L82 0H361L367 28C275 34 266 38 281 121L374 615H441C522 615 543 608 553 583C562 560 566 531 565 493L597 494C603 551 612 629 620 675Z"/></g><g transform="matrix(.013,0,0,-0.013,15.689,0)"><path id="g113-46" d="M170 255C170 288 146 313 114 313S58 288 58 255C58 221 82 198 114 198S170 221 170 255Z"/></g><g transform="matrix(.013,0,0,-0.013,21.558,0)"><use xlink:href="#g113-70"/></g><g transform="matrix(.0091,0,0,-0.0091,29.501,3.132)"><use xlink:href="#g50-114"/></g><g transform="matrix(.013,0,0,-0.013,34.557,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-transform of its Fourier-Laguerre series at frontier point <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M7" height="8.55521pt" version="1.1" viewBox="-0.0498162 -8.34882 28.5806 8.55521" width="28.5806pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-121"/></g><g transform="matrix(.013,0,0,-0.013,10.899,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.013,0,0,-0.013,22.162,0)"><use xlink:href="#g113-49"/></g></svg>, where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M8" height="9.01194pt" version="1.1" viewBox="-0.0498162 -8.96212 8.41168 9.01194" width="8.41168pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-85"/></g></svg> is a general lower triangular regular matrix. A particular case, when <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M9" height="9.01194pt" version="1.1" viewBox="-0.0498162 -8.96212 8.41168 9.01194" width="8.41168pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-85"/></g></svg> is a Ces&#xe1;ro matrix of order 1, that is, <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.1815pt" id="M10" height="11.8174pt" version="1.1" viewBox="-0.0498162 -8.6359 13.6974 11.8174" width="13.6974pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-68"/></g><g transform="matrix(.0091,0,0,-0.0091,8.619,3.132)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g></svg>, has also been discussed as a corollary of main result.

M. L. Mittal

Mradul Veer Singh

Error Estimation of Functions by Fourier-Laguerre Polynomials Using Matrix-Euler Operators

K. Itoh

R. Sakai

N. Suzuki

The de la Vall&#xE9;e Poussin Mean and Polynomial Approximation for Exponential Weight

About this journal

Journal profile

Meet the editorial board

Editor spotlight

Find out more

This journal's articles appear in a wide range of abstracting and indexing databases, and are covered by numerous other services that aid discovery and access. Find out more about where and how the content of this journal is available.

International Journal of Analysis