Some New Inequalities for Lipschitz Functions via a Functional

Year 2019, Volume: 9 Issue: 2, 301 - 306, 15.04.2019

Mahir Kadakal , İmdat İşcan , Cuma Altunsoy

https://doi.org/10.17714/gumusfenbil.436723

Abstract

This study is about getting some new integral
inequalities for Lipschitz functions by using a functional defined via a
Lipschitz function. Here, some new Hermite-Hadamard (H-H) type inequalities are
first found out as a corollary of main theorems. Afterwards, some new H-H type
inequalities for Lipschitz functions by means of inequalities which are used
for -convex functions are obtained.

Keywords

Convex function, Lipschitz function, p-Convex function

Bir Fonksiyonel yardımı ile Lipschitz Fonksiyonları için Bazı Yeni Eşitsizlikler

Abstract

Bu çalışma, bir
Lipschitz fonksiyonu yardımı ile tanımlanmış bir fonksiyonel kullanarak
Lipschitz fonksiyonları için bazı yeni integral eşitsizliklerin elde edilmesi
ile ilgilidir. Burada ilk önce, bazı yeni Hermite-Hadamard tipi eşitsizlikler,
ana teoremlerin bir sonucu olarak ortaya çıkarılacaktır. Daha sonra ise, -konveks
fonksiyonlar için kullanılan eşitsizlikler aracılığıyla Lipschitz fonksiyonları
için yeni Hermite Hadamard tipi eşitsizlikler elde edilecektir.

Keywords

Konveks fonksiyon, Lipschitz fonksiyonu, p-konveks fonksiyon

References

• Dragomir, S.S. Cho, Y.J. and Kim, S.S., 2000, Inequalities of Hadamard's type for Lipschitzian mappings and their applications, Journal of Mathematical Analysis and Applications, vol. 245, no. 2, pp. 489-501.
• Dragomir, S.S., Pearce, C.E.M., 2002, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monograph, Victoria University, online:http://rgmia.org/monographs.php.
• Dragomir, S.S., 2002, On Some New inequalities of Hermite-Hadamrd Type for m-Convex Functions, Tamkang J. of Math., vol. 33, 1, 45-55.
• Hadamard, J., 1893, Etude sur les proprietes des fonctions entieres en particulier d'une fonction consideree par Riemann, J. Math. Pures Appl. 58, 171-215.
• İşcan, İ., 2014, Hermite-Hadamard type inequalities for harmonically convex functions. Hacettepe Journal of Mathematics and Statistics, 43(6), 935-942.
• İşcan, İ., 2016, Ostrowski type inequalities for p-convex functions, New Trends in Mathematical Sciences, NTMSCI 4, No. 3, 140-150.
• İşcan, İ., Altunsoy, C. and Kadakal, M., 2018, New inequalities on Lipschitz functions, International Conference on Mathematics and Mathematics Education, Ordu University, Ordu, 27-29 Haziran, Book of abstracts, s.169.
• Kunt, M. and İşcan, İ., 2017a, On new Hermite-Hadamard-Fejer type inequalities for p-convex functions via fractional integrals, Communication in Mathematical Modeling and Applications, Volume: 2, Issue: 1, June, pp:1-15.
• Kunt, M. and İşcan, İ., 2017b, Hermite-Hadamard-Fejer type inequalities for p-convex functions, Arab J. Math. Sci., 23(2), 215-230.
• Kunt, M. and İşcan, İ., 2017c, Hermite-Hadamard-Fejer type inequalities for p-convex functions via fractional integrals, Iranian Journal of Science and Technology, Transactions A: Science, Doi:10.1007/s40995-017-0352-4.
• Kunt, M. and İşcan, İ., 2017d, Hermite-Hadamard type inequalities for p-convex functions via fractional integrals, Moroccan J. Pure Appl. Anal., 3(1), 22-35.
• Latif, M.A. Dragomir S. S. and Momaniat, E., 2015, Some Fejer type integral inequalities for geometrically-arithmetically-convex functions with applications, RGMIA Research Report Collection, 18, Article 25, 18 pp.
• Niculescu, C.P., 2000, Convexity according to the geometric mean. Math. Inequal. Appl., 3(2):155-167. 10.7153/mia-03-19.
• Pečarić, J., Proschan, F. and Tong, Y. L., 1992, Convex Functions, Partial Orderings and Statistical Applications. Academic Press, Inc., 469 pp, Boston.
• Roberts, A.W. and Varberg, D.E., 1973, Convex Functions. Academic Press, 300 pp, New York.
• Yang, G.S. and Tseng, K.L., 1999, On certain integral inequalities related to Hermite-Hadamard inequalities, J. Math. Anal. Appl., 239, 180-187.