Araştırma Makalesi
BibTex RIS Kaynak Göster

The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function

Yıl 2025, Cilt: 54 Sayı: 2, 404 - 413, 28.04.2025

Gültekin Tınaztepe , Sinem Sezer Evcan , Zeynep Eken , Sevda Sezer

https://doi.org/10.15672/hujms.1444589

Öz

By means of an integral identity, several Hermite-Hadamard type inequalities are presented in this study for a function whose derivative's absolute value is the log-p-convex function. With the use of these findings, we are able to determine the boundaries in terms of elementary functions for certain specific functions, such as the imaginary error function, the exponential integral, the hyperbolic sine and cosine functions. Additionally, a relationship between beta function, the hyperbolic sine and cosine functions is stated. Through the obtained results, a bound for numerical integration of such type functions is provided.

Anahtar Kelimeler

p-convex set log p-convex function p-convex function Hermite-Hadamard inequality

Kaynakça

  • [1] G. Adilov and I. Yesilce, $B^{-1}$-Convex Functions, J. Convex Anal. 24 (2), 505-517, 2017.
  • [2] H. Boche and M. Schubert, A Calculus for log-Convex Interference Functions, IEEE Transactions on Information Theory 54 (12), 5469-5490, 2008.
  • [3] Desmos Graphing Calculator. 2023. Desmos Graphing Calculator. [online] Available at: < https://www.desmos.com/calculator/frvsxdcdwg > [Accessed 9 April 2023]
  • [4] Desmos Graphing Calculator. 2023. Desmos Graphing Calculator. [online] Available at: < https://www.desmos.com/calculator/tpi5wutstg > [Accessed 9 April 2023]
  • [5] S. S. Dragomir and C. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, Math. Prep. Archive, 3, 463-817, 2003.
  • [6] Z. Eken, S. Sezer, G. Tınaztepe and G. Adilov, s-Convex Functions in the Fourth Sense and Some of Their Properties, Konuralp J. Math. 9 (2), 260-267, 2021.
  • [7] S. Kemali, Hermite-Hadamard Type Inequality For s-Convex Functions in the Fourth Sense, Turkish J. Math. Comp. Sci. 13 (2), 287-293, 2021.
  • [8] S. Kemali, S. Sezer, G. Tınaztepe and G. Adilov, s-Convex Functions in the Third Sense. Korean J. Math. 29 (3) , 593-602, 2021.
  • [9] S. Kemali, S. Sezer Evcan, I. Yesilce Isik and G. Adilov, Some Integral Inequalities for the Product of s-Convex Functions in the Fourth Sense. Int. J. Nonlinear Anal. Appl. 13 (2), 103-116, 2022.
  • [10] S. Kemali, I. Yesilce and G. Adilov, $\mathbb{B}$ -Convexity, $\mathbb{B}^{-1}$-Convexity, And Their Comparison, Numer. Funct. Anal. Optim. 36 (2), 133-146, 2015.
  • [11] A. Klinger and O. L. Mangasarian, Logarithmic Convexity and Geometric Programming, J. Mat. Anal. Appl. 24 (2), 388-408, 1968.
  • [12] R. J. Knops, Logarithmic Convexity and Other Techniques Applied to Problems in Continuum Mechanics. In Symposium on Non-Well-Posed Problems and Logarithmic Convexity: Held in Heriot-Watt University, Edinburgh/Scotland March 22-24, 1972 (31-54), Springer Berlin Heidelberg, 2006.
  • [13] H. Mamedov and I. Yesilce Isik, On the Fractional Integral Inequalities for p-convex Functions, Punjab Univ. J. Math. 55 (5-6), 185-196, 2023.
  • [14] M. A. Noor and K. I. Noor, New Perspectives of log-Convex Functions. Applied Math. Inf. Sci. 14 (5), 847-854, 2020.
  • [15] M. A. Noor and K. I. Noor, Strongly log-Convex Functions. Information Sciences Letters 10 (1), 33-38, 2021.
  • [16] R. Quintanilla, On the Logarithmic Convexity in Thermoelasticity with Microtemperatures. Journal of Thermal Stresses. 36 (4), 378-386, 2013.
  • [17] C. P. Rydell, The Significance of Logarithmic Convexity for Price and Growth Theory, Western Economic Journal Oxford, 6 (1), 65-71, 1967.
  • [18] S. Sezer, Hermite-Hadamard Type Inequalities for the Functions Whose Absolute Values of First Derivatives are p-Convex. Fundamental Journal of Mathematics and Applications, 4 (2), 88-99, 2021.
  • [19] S. Sezer, Z. Eken, G. Tınaztepe and G. Adilov, p-Convex Functions and Their Some Properties, Numer. Funct. Anal. Optim. 42 (4), 443-459, 2021.
  • [20] G. Tınaztepe, S. Sezer, Z. Eken and G. Adilov. Quasi p-Convex Functions. Appl. Math. E-Notes, 22, 741-750, 2022.
  • [21] G. Tınaztepe, S. Sezer, Z. Eken and G. Adilov, Logarithmic p-Convex Functions and Some of Their Properties, 2024, (submitted).
  • [22] I. Yesilce and G. Adilov, Hermite-Hadamard Inequalities For L (J)-Convex Functions And S (J)-Convex Functions, Malaya Journal of Matematik, 3 (3), 346-359, 2015.
  • [23] I. Yesilce and G. Adilov, Hermite-Hadamard Inequalities for $\mathbb{B}$-Convex and $\mathbb{B}^{-1}$- Convex Functions. International Journal of Nonlinear Analysis and Applications, 8 (1), 225-233, 2017.

Yıl 2025, Cilt: 54 Sayı: 2, 404 - 413, 28.04.2025

Gültekin Tınaztepe , Sinem Sezer Evcan , Zeynep Eken , Sevda Sezer

https://doi.org/10.15672/hujms.1444589

Öz

Kaynakça

  • [1] G. Adilov and I. Yesilce, $B^{-1}$-Convex Functions, J. Convex Anal. 24 (2), 505-517, 2017.
  • [2] H. Boche and M. Schubert, A Calculus for log-Convex Interference Functions, IEEE Transactions on Information Theory 54 (12), 5469-5490, 2008.
  • [3] Desmos Graphing Calculator. 2023. Desmos Graphing Calculator. [online] Available at: < https://www.desmos.com/calculator/frvsxdcdwg > [Accessed 9 April 2023]
  • [4] Desmos Graphing Calculator. 2023. Desmos Graphing Calculator. [online] Available at: < https://www.desmos.com/calculator/tpi5wutstg > [Accessed 9 April 2023]
  • [5] S. S. Dragomir and C. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, Math. Prep. Archive, 3, 463-817, 2003.
  • [6] Z. Eken, S. Sezer, G. Tınaztepe and G. Adilov, s-Convex Functions in the Fourth Sense and Some of Their Properties, Konuralp J. Math. 9 (2), 260-267, 2021.
  • [7] S. Kemali, Hermite-Hadamard Type Inequality For s-Convex Functions in the Fourth Sense, Turkish J. Math. Comp. Sci. 13 (2), 287-293, 2021.
  • [8] S. Kemali, S. Sezer, G. Tınaztepe and G. Adilov, s-Convex Functions in the Third Sense. Korean J. Math. 29 (3) , 593-602, 2021.
  • [9] S. Kemali, S. Sezer Evcan, I. Yesilce Isik and G. Adilov, Some Integral Inequalities for the Product of s-Convex Functions in the Fourth Sense. Int. J. Nonlinear Anal. Appl. 13 (2), 103-116, 2022.
  • [10] S. Kemali, I. Yesilce and G. Adilov, $\mathbb{B}$ -Convexity, $\mathbb{B}^{-1}$-Convexity, And Their Comparison, Numer. Funct. Anal. Optim. 36 (2), 133-146, 2015.
  • [11] A. Klinger and O. L. Mangasarian, Logarithmic Convexity and Geometric Programming, J. Mat. Anal. Appl. 24 (2), 388-408, 1968.
  • [12] R. J. Knops, Logarithmic Convexity and Other Techniques Applied to Problems in Continuum Mechanics. In Symposium on Non-Well-Posed Problems and Logarithmic Convexity: Held in Heriot-Watt University, Edinburgh/Scotland March 22-24, 1972 (31-54), Springer Berlin Heidelberg, 2006.
  • [13] H. Mamedov and I. Yesilce Isik, On the Fractional Integral Inequalities for p-convex Functions, Punjab Univ. J. Math. 55 (5-6), 185-196, 2023.
  • [14] M. A. Noor and K. I. Noor, New Perspectives of log-Convex Functions. Applied Math. Inf. Sci. 14 (5), 847-854, 2020.
  • [15] M. A. Noor and K. I. Noor, Strongly log-Convex Functions. Information Sciences Letters 10 (1), 33-38, 2021.
  • [16] R. Quintanilla, On the Logarithmic Convexity in Thermoelasticity with Microtemperatures. Journal of Thermal Stresses. 36 (4), 378-386, 2013.
  • [17] C. P. Rydell, The Significance of Logarithmic Convexity for Price and Growth Theory, Western Economic Journal Oxford, 6 (1), 65-71, 1967.
  • [18] S. Sezer, Hermite-Hadamard Type Inequalities for the Functions Whose Absolute Values of First Derivatives are p-Convex. Fundamental Journal of Mathematics and Applications, 4 (2), 88-99, 2021.
  • [19] S. Sezer, Z. Eken, G. Tınaztepe and G. Adilov, p-Convex Functions and Their Some Properties, Numer. Funct. Anal. Optim. 42 (4), 443-459, 2021.
  • [20] G. Tınaztepe, S. Sezer, Z. Eken and G. Adilov. Quasi p-Convex Functions. Appl. Math. E-Notes, 22, 741-750, 2022.
  • [21] G. Tınaztepe, S. Sezer, Z. Eken and G. Adilov, Logarithmic p-Convex Functions and Some of Their Properties, 2024, (submitted).
  • [22] I. Yesilce and G. Adilov, Hermite-Hadamard Inequalities For L (J)-Convex Functions And S (J)-Convex Functions, Malaya Journal of Matematik, 3 (3), 346-359, 2015.
  • [23] I. Yesilce and G. Adilov, Hermite-Hadamard Inequalities for $\mathbb{B}$-Convex and $\mathbb{B}^{-1}$- Convex Functions. International Journal of Nonlinear Analysis and Applications, 8 (1), 225-233, 2017.
Toplam 23 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Uygulamalı Matematik (Diğer)
Bölüm Matematik
Yazarlar

Gültekin Tınaztepe 0000-0001-7594-1620

Sinem Sezer Evcan 0000-0003-2066-7833

Zeynep Eken 0000-0002-8939-4653

Sevda Sezer 0000-0001-6448-193X

Erken Görünüm Tarihi 27 Ağustos 2024
Yayımlanma Tarihi 28 Nisan 2025
Gönderilme Tarihi 28 Şubat 2024
Kabul Tarihi 23 Nisan 2024
Yayımlandığı Sayı Yıl 2025 Cilt: 54 Sayı: 2

Kaynak Göster

APA Tınaztepe, G., Sezer Evcan, S., Eken, Z., Sezer, S. (2025). The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function. Hacettepe Journal of Mathematics and Statistics, 54(2), 404-413. https://doi.org/10.15672/hujms.1444589
AMA Tınaztepe G, Sezer Evcan S, Eken Z, Sezer S. The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function. Hacettepe Journal of Mathematics and Statistics. Nisan 2025;54(2):404-413. doi:10.15672/hujms.1444589
Chicago Tınaztepe, Gültekin, Sinem Sezer Evcan, Zeynep Eken, ve Sevda Sezer. “The Hermite-Hadamard Type Inequalities for the Functions Whose Derivative Is Logarithmic $p$-Convex Function”. Hacettepe Journal of Mathematics and Statistics 54, sy. 2 (Nisan 2025): 404-13. https://doi.org/10.15672/hujms.1444589.
EndNote Tınaztepe G, Sezer Evcan S, Eken Z, Sezer S (01 Nisan 2025) The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function. Hacettepe Journal of Mathematics and Statistics 54 2 404–413.
IEEE G. Tınaztepe, S. Sezer Evcan, Z. Eken, ve S. Sezer, “The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function”, Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 2, ss. 404–413, 2025, doi: 10.15672/hujms.1444589.
ISNAD Tınaztepe, Gültekin vd. “The Hermite-Hadamard Type Inequalities for the Functions Whose Derivative Is Logarithmic $p$-Convex Function”. Hacettepe Journal of Mathematics and Statistics 54/2 (Nisan 2025), 404-413. https://doi.org/10.15672/hujms.1444589.
JAMA Tınaztepe G, Sezer Evcan S, Eken Z, Sezer S. The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function. Hacettepe Journal of Mathematics and Statistics. 2025;54:404–413.
MLA Tınaztepe, Gültekin vd. “The Hermite-Hadamard Type Inequalities for the Functions Whose Derivative Is Logarithmic $p$-Convex Function”. Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 2, 2025, ss. 404-13, doi:10.15672/hujms.1444589.
Vancouver Tınaztepe G, Sezer Evcan S, Eken Z, Sezer S. The Hermite-Hadamard type inequalities for the functions whose derivative is logarithmic $p$-convex function. Hacettepe Journal of Mathematics and Statistics. 2025;54(2):404-13.

For more information about the journal, please visit: https://dergipark.org.tr/en/pub/hujms