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On Some Cauchy Type Mean-Value Theorems with Applications

Yıl 2024, Cilt: 7 Sayı: 3, 147 - 156, 29.09.2024

Uğur Selamet Kırmacı

https://doi.org/10.33434/cams.1503610
Cited By: 1

Öz

Some Cauchy-type mean-value theorems for Chebychev’s inequality, Steffensen’s inequality, midpoint rule, and Simpson’s rule are presented. Furthermore, we give some applications for the obtained results using the exponential and logarithmic functions, their Taylor polynomials, and some trigonometric functions. Further, we obtain some exponential, logarithmic, and trigonometric equations and give two inequalities for midpoint and Simpson’s rules.

Anahtar Kelimeler

Cauchy mean-value theorem Chebychev’s inequality Midpoint rule Steffensen’s inequality Simpson’s rule

Kaynakça

  • [1] U. Abel, M. Ivan, T. Riedel, The mean-value theorem of flett and divided differences, J. Math. Anal. Appl., 295 (2004), 1-9.
  • [2] G. Farid, M. Marwan, A. Ur Rehman, New mean-value theorems and generalization of Hadamard inequality via coordinated m-convex functions, J. Inequal. Appl., 283 (2015), 1-11.
  • [3] J. Matkowski, A mean-value theorem and its applications, J. Math. Anal. Appl., 373 (2011), 227–234.
  • [4] J. A. Reyna, A generalized mean-value theorem, Mh. Math., 106 (1988), 95-97.
  • [5] A. McD. Mercer, Some new inequalities involving elementary mean values, J. Math. Anal. Appl., 229 (1999), 677-681.
  • [6] J. E. Pecaric, I. Peric, H. M. Srivastava, A family of the Cauchy type mean-value Theorems, J. Math. Anal. Appl., 306 (2005), 730-739.
  • [7] C. E. M. Pearce, Stolarsky means and Hadamard’s inequality, J. Math. Anal. Appl., 220 (1998), 99-109.
  • [8] F. Qi, Generalized abstracted mean values, J. Inequal. Pure Appl. Math., 1(1) (2000), Article 4, 9 pages.
  • [9] M. Anwar, N. Latif, J.E. Peˇcari´c, Cauchy means of the Popoviciu type, J. Inequal. Appl., (2009), Article ID 628051, 16 pages.
  • [10] S. Abramovich, G. Farid, J. E. Peˇcari´c, More about Hermite-Hadamard Inequalities, Cauchy’s means, and superquadracity, J. Inequal. Appl., (2010), Article ID 102467, 14 pages.
  • [11] N. Mehreen, M. Anwar, Cauchy type means for some generalized convex functions, J. Inequal. Appl., 114 (2021), 1-13.
  • [12] L. Horvath, K. Ali Khan, J. E. Pecaric, Refinements of results about weighted mixed symmetric means and related Cauchy means, J. Inequal. Appl., (2011), Article ID 350973, 19 pages.
  • [13] M. Anwar, N. Latif, J.E. Peˇcari´c, Positive semidefinite matrices, exponential convexity for majorization, and related Cauchy means, J. Inequal. Appl., (2010), Article ID 728251, 19 pages.
  • [14] J.E. Peˇcari´c, M. R. Lipanovi´c, H. M. Srivastava, Some mean-value theorems of the Cauchy type, Fract. Calc. Appl. Anal., 9(2) (2006), 143-158.
  • [15] D. Andrica, T. M. Rassias, Differential and Integral Inequalities, Springer Optimization and Its Applications, 151, 2019.
  • [16] D. S. Mitrinovic, Analytic Inequalities, Springer-VerlagNew-York, Heidelberg, Berlin, 1970.
  • [17] M. Alomari, M. Darus, U. S.Kirmaci, Refinements of Hadamard-type inequalities for quasi-convex functions with applictions to trapezoidal formula and to special means, Comput. Math. Appl., 59 (1) (2010), 225-232.
  • [18] O. Hutnik, On Hadamard type inequalities for generalized weighted quasi-arithmetic means, J. Inequal. Pure Appl. Math.,7(3) (2006), Article 96, 10 pages.
  • [19] U. S. Kirmaci, On new integral inequalities with applications, American Review of Mathematics and Statistics, 2(2) (2014),107-124.
  • [20] U. S. Kirmaci, M. K. Bakula, M. E.O¨ zdemir, J. E. Pecˇaric´, On some inequalities for p-norms, J. Inequal. Pure Appl. Math.,9(1) (2008), Article 27, 8 pages.
  • [21] U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 147 (2004), 137-146.
  • [22] U. S. Kirmaci, R. Dikici, On some Hermite-Hadamard type inequalities for twice differentiable mappings and applications, Tamkang J. Math., 44(1) (2013), 41-51.
  • [23] U.S. Kirmaci, M. K. Bakula, M.E. Özdemir, J.E. Pecaric, Hadamard-type inequalities for s-convex functions, Appl. Math. Comput., 193 (2007), 26–35.
  • [24] U.S. Kirmaci, Improvement and further generalization of inequalities for differentiable mappings and applications, Comput. Math. Appl., 55 (2008), 485–493.
  • [25] U. S. Kirmaci, M. E. Özdemir, On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 153(2004), 361–368.
  • [26] U. S. Kirmaci, M. E. Özdemir, Some inequalities for mappings whose derivatives are bounded and applications to special means of real numbers, Appl. Math. Lett., 17(2004), 641–645.
  • [27] U. S. Kirmaci, Refinements of Hermite-Hadamard type inequalities for s-convex functions with applications to special means, Univers. J. Math. Appl., 4(3) (2021), 114-124.
  • [28] U. S. Kirmaci, On generalizations of Holder’s and Minkowski’s inequalities, Math. Sci. Appl. E-Notes, 11 (2023), 213–225.
  • [29] S. S. Dragomir, Ostrowski and trapezoid type inequalities for the generalized k-g-fractional integrals of functions with bounded variation, Commun. Adv. Math. Sci., II(4) (2019), 309-330.
  • [30] M. Tunc, On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat, 27(4) (2013), 559–565.

Yıl 2024, Cilt: 7 Sayı: 3, 147 - 156, 29.09.2024

Uğur Selamet Kırmacı

https://doi.org/10.33434/cams.1503610
Cited By: 1

Öz

Kaynakça

  • [1] U. Abel, M. Ivan, T. Riedel, The mean-value theorem of flett and divided differences, J. Math. Anal. Appl., 295 (2004), 1-9.
  • [2] G. Farid, M. Marwan, A. Ur Rehman, New mean-value theorems and generalization of Hadamard inequality via coordinated m-convex functions, J. Inequal. Appl., 283 (2015), 1-11.
  • [3] J. Matkowski, A mean-value theorem and its applications, J. Math. Anal. Appl., 373 (2011), 227–234.
  • [4] J. A. Reyna, A generalized mean-value theorem, Mh. Math., 106 (1988), 95-97.
  • [5] A. McD. Mercer, Some new inequalities involving elementary mean values, J. Math. Anal. Appl., 229 (1999), 677-681.
  • [6] J. E. Pecaric, I. Peric, H. M. Srivastava, A family of the Cauchy type mean-value Theorems, J. Math. Anal. Appl., 306 (2005), 730-739.
  • [7] C. E. M. Pearce, Stolarsky means and Hadamard’s inequality, J. Math. Anal. Appl., 220 (1998), 99-109.
  • [8] F. Qi, Generalized abstracted mean values, J. Inequal. Pure Appl. Math., 1(1) (2000), Article 4, 9 pages.
  • [9] M. Anwar, N. Latif, J.E. Peˇcari´c, Cauchy means of the Popoviciu type, J. Inequal. Appl., (2009), Article ID 628051, 16 pages.
  • [10] S. Abramovich, G. Farid, J. E. Peˇcari´c, More about Hermite-Hadamard Inequalities, Cauchy’s means, and superquadracity, J. Inequal. Appl., (2010), Article ID 102467, 14 pages.
  • [11] N. Mehreen, M. Anwar, Cauchy type means for some generalized convex functions, J. Inequal. Appl., 114 (2021), 1-13.
  • [12] L. Horvath, K. Ali Khan, J. E. Pecaric, Refinements of results about weighted mixed symmetric means and related Cauchy means, J. Inequal. Appl., (2011), Article ID 350973, 19 pages.
  • [13] M. Anwar, N. Latif, J.E. Peˇcari´c, Positive semidefinite matrices, exponential convexity for majorization, and related Cauchy means, J. Inequal. Appl., (2010), Article ID 728251, 19 pages.
  • [14] J.E. Peˇcari´c, M. R. Lipanovi´c, H. M. Srivastava, Some mean-value theorems of the Cauchy type, Fract. Calc. Appl. Anal., 9(2) (2006), 143-158.
  • [15] D. Andrica, T. M. Rassias, Differential and Integral Inequalities, Springer Optimization and Its Applications, 151, 2019.
  • [16] D. S. Mitrinovic, Analytic Inequalities, Springer-VerlagNew-York, Heidelberg, Berlin, 1970.
  • [17] M. Alomari, M. Darus, U. S.Kirmaci, Refinements of Hadamard-type inequalities for quasi-convex functions with applictions to trapezoidal formula and to special means, Comput. Math. Appl., 59 (1) (2010), 225-232.
  • [18] O. Hutnik, On Hadamard type inequalities for generalized weighted quasi-arithmetic means, J. Inequal. Pure Appl. Math.,7(3) (2006), Article 96, 10 pages.
  • [19] U. S. Kirmaci, On new integral inequalities with applications, American Review of Mathematics and Statistics, 2(2) (2014),107-124.
  • [20] U. S. Kirmaci, M. K. Bakula, M. E.O¨ zdemir, J. E. Pecˇaric´, On some inequalities for p-norms, J. Inequal. Pure Appl. Math.,9(1) (2008), Article 27, 8 pages.
  • [21] U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 147 (2004), 137-146.
  • [22] U. S. Kirmaci, R. Dikici, On some Hermite-Hadamard type inequalities for twice differentiable mappings and applications, Tamkang J. Math., 44(1) (2013), 41-51.
  • [23] U.S. Kirmaci, M. K. Bakula, M.E. Özdemir, J.E. Pecaric, Hadamard-type inequalities for s-convex functions, Appl. Math. Comput., 193 (2007), 26–35.
  • [24] U.S. Kirmaci, Improvement and further generalization of inequalities for differentiable mappings and applications, Comput. Math. Appl., 55 (2008), 485–493.
  • [25] U. S. Kirmaci, M. E. Özdemir, On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 153(2004), 361–368.
  • [26] U. S. Kirmaci, M. E. Özdemir, Some inequalities for mappings whose derivatives are bounded and applications to special means of real numbers, Appl. Math. Lett., 17(2004), 641–645.
  • [27] U. S. Kirmaci, Refinements of Hermite-Hadamard type inequalities for s-convex functions with applications to special means, Univers. J. Math. Appl., 4(3) (2021), 114-124.
  • [28] U. S. Kirmaci, On generalizations of Holder’s and Minkowski’s inequalities, Math. Sci. Appl. E-Notes, 11 (2023), 213–225.
  • [29] S. S. Dragomir, Ostrowski and trapezoid type inequalities for the generalized k-g-fractional integrals of functions with bounded variation, Commun. Adv. Math. Sci., II(4) (2019), 309-330.
  • [30] M. Tunc, On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat, 27(4) (2013), 559–565.
Toplam 30 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Temel Matematik (Diğer)
Bölüm Makaleler
Yazarlar

Uğur Selamet Kırmacı 0000-0002-8177-6649

Erken Görünüm Tarihi 8 Eylül 2024
Yayımlanma Tarihi 29 Eylül 2024
Gönderilme Tarihi 23 Haziran 2024
Kabul Tarihi 6 Eylül 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 7 Sayı: 3

Kaynak Göster

APA Kırmacı, U. S. (2024). On Some Cauchy Type Mean-Value Theorems with Applications. Communications in Advanced Mathematical Sciences, 7(3), 147-156. https://doi.org/10.33434/cams.1503610
AMA Kırmacı US. On Some Cauchy Type Mean-Value Theorems with Applications. Communications in Advanced Mathematical Sciences. Eylül 2024;7(3):147-156. doi:10.33434/cams.1503610
Chicago Kırmacı, Uğur Selamet. “On Some Cauchy Type Mean-Value Theorems With Applications”. Communications in Advanced Mathematical Sciences 7, sy. 3 (Eylül 2024): 147-56. https://doi.org/10.33434/cams.1503610.
EndNote Kırmacı US (01 Eylül 2024) On Some Cauchy Type Mean-Value Theorems with Applications. Communications in Advanced Mathematical Sciences 7 3 147–156.
IEEE U. S. Kırmacı, “On Some Cauchy Type Mean-Value Theorems with Applications”, Communications in Advanced Mathematical Sciences, c. 7, sy. 3, ss. 147–156, 2024, doi: 10.33434/cams.1503610.
ISNAD Kırmacı, Uğur Selamet. “On Some Cauchy Type Mean-Value Theorems With Applications”. Communications in Advanced Mathematical Sciences 7/3 (Eylül 2024), 147-156. https://doi.org/10.33434/cams.1503610.
JAMA Kırmacı US. On Some Cauchy Type Mean-Value Theorems with Applications. Communications in Advanced Mathematical Sciences. 2024;7:147–156.
MLA Kırmacı, Uğur Selamet. “On Some Cauchy Type Mean-Value Theorems With Applications”. Communications in Advanced Mathematical Sciences, c. 7, sy. 3, 2024, ss. 147-56, doi:10.33434/cams.1503610.
Vancouver Kırmacı US. On Some Cauchy Type Mean-Value Theorems with Applications. Communications in Advanced Mathematical Sciences. 2024;7(3):147-56.

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https://doi.org/10.36753/mathenot.1500238
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