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Çok Boyutlu Fonksiyonların Genelleştirilmiş Konveksliği ve İlgili Hermite-Hadamard Tipi İntegral Eşitsizlikleri

Year 2020, , 241 - 250, 15.01.2020
https://doi.org/10.17714/gumusfenbil.632639

Abstract

Bu makalede temel amaç, çok boyutlu fonksiyonların genelleştirilmiş konveksliğini incelemek ve onunla ilgili bazı önemli eşitsizlikler elde etmektir. Bu nedenle ilk olarak çok boyutlu genelleştirilmiş konveks fonksiyonlar tanımlanmıştır. Devamında bu fonksiyonların bazı özelliklerinden bahsedilmiştir. Buna bağlı olarak, çok boyutlu genelleştirilmiş konveks fonksiyonlar ile diğer konveks fonksiyonların ilişkisi kurulmuştur. Ek olarak, iki boyutlu genelleştirilmiş konveks fonksiyonlar için Hermite-Hadamard tipli integral eşitsizliği genelleştirilmiştir. Son olarak bu çalışmada, çok boyutlu genelleştirilmiş konveks için Hermite-Hadamard tipli integral eşitsizliği elde edilmiş ve bu eşitsizliği açıklayıcı bir örnek verilmiştir

References

  • Agarwal, R.P., Luo, M.-J., and Raina, R.K., 2016. On Ostrowski type inequalities. Fasciculi Mathematici, 204, 5-27.
  • Cristescu, G., 2004. Hadamard type inequalities for φ-convex functions: Annals of the University of Oradea, Fascicle of Management and Technological Engineering, CD-Rom Edition, III(XIII).
  • De la Cal, J. and Carcamo, J., 2006. Multidimensional Hermite-Hadamard inequalities and the convex order. Journal of Math. Analysis and Applications, 324, 248-261.
  • Dragomir, S.S., 2015. Inequalities of Jensen Type for φ-Convex Functions. Fasciculi Mathematici, 5, 35-52.
  • Dragomir, S.S. and Pearce, C.E.M., 2000. Selected Topics on Hermite-Hadamard Inequalities and Appications. RGMIA Monographs, Victoria University.
  • Dragomir, S.S., 1992. Two Mappings in Connection to Hadamard’s Inequalities. Journal of Mathematıcal Analysis and Applications 167, 49-56.
  • Dragomir, S.S, 2001. On Hadamards inequality for convex functions on the coordinates in a rectangle from the plane. Taiwanese J. Math., 4, 775–788.
  • Ellahi, H., Farid, G., and Rehman, A.U., 2015. Hadamard’s Inequality for s-convex function on n-coordinates, Proceedings of 1st ICAM Attock, Pakistan . Hadamard, J., 1893. Étude sur les propriétés des fonctions entières et en particulier d.une function considerée par Riemann. J. Math Pures Appl., 58, 171-215.
  • Latif M.A. and Alomari, M., 2009. On Hadamard-type inequalities for h−convex functions on the co-ordinates. Int. Journal of Math. Analysis, 33, 1645-1656.
  • Martinez-Legaz, J.E. and Singer, I., 1998. On φ-convexity of convex functions. Linear Algebra and its Applications, 278, 163-181.
  • Okur, N., İşcan, İ., and Usta, Y., 2018. Some Integral Inequalities for Harmonically Convex Stochastic Processes on the Coordinates. Advanced Math. Models & Applications, 3(1), 63-75.
  • Okur, N., İşcan, İ., and Yüksek Dizdar, E., 2019. Hermite-Hadamard Type Inequalities for p-Convex Stochastic Processes. An International Journal of Optimization and Control: Theories & Applications, 9(2), 148-153.
  • Okur, N., 2019. Multidimensional General Convexity for Stochastic Processes and Associated with Hermite-Hadamard Type Integral Inequalities. Thermal Science, OnLine-First (00):361-361, https://doi.org/10.2298/TSCI190622361O.
  • Okur, N. and Karahan, V., 2019. Some Integral Inequalities of the Hermite-Hadamard Type for s-Convex Stochastic Processes on n-coordinates. Commun. Fac. Sci. Univ. Ank. Ser. A1, 68(2), 1959-1973.
  • Okur, N. and Yalçın, F.B., 2019. Two-Dimensional Operator Harmonically Convex Functions and Related Generalized Inequalities. Turkish Journal of Science, 5(1), 30-38.
  • Sarikaya, M.Z., Büyükeken, M., and Kiriş, M.E., 2015. On Some Generalized Integral Inequalities for φ-Convex Functions, Studia Univ. Babeş-Bolyai Mathematica, 60(3), 367–377.
  • Sarikaya, M.Z., Kiriş, M.E. and Çelik, N., 2016. Hermite-Hadamard Type Inequality for φ_h-Convex Functions. AIP Conference Proceedings 1726, 020076, doi: 10.1063/1.4945902.
  • Set, E., Sarıkaya, M. Z., and Akdemir, A. O., 2014. Hadamard type inequalities for φ−convex functions on the co-ordinates. Tbilisi Mathematical Journal, 7(2), 51–60.
  • Shaikh A.A., Iqbal, A., and Mondal, C.K., 2018. Some Results on φ-Convex Functions and Geodesic Φ-Convex Functions. Differential Geometry-Dynamical Systems, 20, 159-169.
  • Syau, Y.R. and Lee, E.S., 2005. Some Properties of E-Convex Functions. Appl. Math. Lett., 18, 1074–1080.
  • Viloria, J.M. and Cortez, M.V., 2018. Hermite-Hadamard type inequalities for harmonically convex functions on n-coordinates. Appl. Math. Inf. Sci. Lett., 6, 1-6.
  • Yalçın, F.B., 2019. On Some Generalized Inequalities for Two-Dimensional 𝜑-Convex Functions, Journal of Contemporary Applied Mathematics, 9(1), 88-102.
  • Youness, E.A., 1999. E-Convex Sets, E-Convex Functions and E-Convex Programming. J. Optimiz. Theory.App., 102, 439–450.

General Convexity of Multidimensional Functions and Related Hermite-Hadamard Type Integral Inequalities

Year 2020, , 241 - 250, 15.01.2020
https://doi.org/10.17714/gumusfenbil.632639

Abstract

The basic goal is to investigate general convexity of multidimensional functions and derive several important inequalities associated with it’s in this paper. For this reason, multidimensional general convex functions were firstly defined. Afterwards, some properties of these functions were mentioned. Accordingly, the relation of multidimensional general convex functions with other convex functions was established. Additionally, a generalization of Hermite-Hadamard type integral inequality was showed for two-dimensional general convex functions. Finally, Hermite-Hadamard type integral inequality for multidimensional general convex functions was verified and an explanatory example for this inequality was given in this study.




 

References

  • Agarwal, R.P., Luo, M.-J., and Raina, R.K., 2016. On Ostrowski type inequalities. Fasciculi Mathematici, 204, 5-27.
  • Cristescu, G., 2004. Hadamard type inequalities for φ-convex functions: Annals of the University of Oradea, Fascicle of Management and Technological Engineering, CD-Rom Edition, III(XIII).
  • De la Cal, J. and Carcamo, J., 2006. Multidimensional Hermite-Hadamard inequalities and the convex order. Journal of Math. Analysis and Applications, 324, 248-261.
  • Dragomir, S.S., 2015. Inequalities of Jensen Type for φ-Convex Functions. Fasciculi Mathematici, 5, 35-52.
  • Dragomir, S.S. and Pearce, C.E.M., 2000. Selected Topics on Hermite-Hadamard Inequalities and Appications. RGMIA Monographs, Victoria University.
  • Dragomir, S.S., 1992. Two Mappings in Connection to Hadamard’s Inequalities. Journal of Mathematıcal Analysis and Applications 167, 49-56.
  • Dragomir, S.S, 2001. On Hadamards inequality for convex functions on the coordinates in a rectangle from the plane. Taiwanese J. Math., 4, 775–788.
  • Ellahi, H., Farid, G., and Rehman, A.U., 2015. Hadamard’s Inequality for s-convex function on n-coordinates, Proceedings of 1st ICAM Attock, Pakistan . Hadamard, J., 1893. Étude sur les propriétés des fonctions entières et en particulier d.une function considerée par Riemann. J. Math Pures Appl., 58, 171-215.
  • Latif M.A. and Alomari, M., 2009. On Hadamard-type inequalities for h−convex functions on the co-ordinates. Int. Journal of Math. Analysis, 33, 1645-1656.
  • Martinez-Legaz, J.E. and Singer, I., 1998. On φ-convexity of convex functions. Linear Algebra and its Applications, 278, 163-181.
  • Okur, N., İşcan, İ., and Usta, Y., 2018. Some Integral Inequalities for Harmonically Convex Stochastic Processes on the Coordinates. Advanced Math. Models & Applications, 3(1), 63-75.
  • Okur, N., İşcan, İ., and Yüksek Dizdar, E., 2019. Hermite-Hadamard Type Inequalities for p-Convex Stochastic Processes. An International Journal of Optimization and Control: Theories & Applications, 9(2), 148-153.
  • Okur, N., 2019. Multidimensional General Convexity for Stochastic Processes and Associated with Hermite-Hadamard Type Integral Inequalities. Thermal Science, OnLine-First (00):361-361, https://doi.org/10.2298/TSCI190622361O.
  • Okur, N. and Karahan, V., 2019. Some Integral Inequalities of the Hermite-Hadamard Type for s-Convex Stochastic Processes on n-coordinates. Commun. Fac. Sci. Univ. Ank. Ser. A1, 68(2), 1959-1973.
  • Okur, N. and Yalçın, F.B., 2019. Two-Dimensional Operator Harmonically Convex Functions and Related Generalized Inequalities. Turkish Journal of Science, 5(1), 30-38.
  • Sarikaya, M.Z., Büyükeken, M., and Kiriş, M.E., 2015. On Some Generalized Integral Inequalities for φ-Convex Functions, Studia Univ. Babeş-Bolyai Mathematica, 60(3), 367–377.
  • Sarikaya, M.Z., Kiriş, M.E. and Çelik, N., 2016. Hermite-Hadamard Type Inequality for φ_h-Convex Functions. AIP Conference Proceedings 1726, 020076, doi: 10.1063/1.4945902.
  • Set, E., Sarıkaya, M. Z., and Akdemir, A. O., 2014. Hadamard type inequalities for φ−convex functions on the co-ordinates. Tbilisi Mathematical Journal, 7(2), 51–60.
  • Shaikh A.A., Iqbal, A., and Mondal, C.K., 2018. Some Results on φ-Convex Functions and Geodesic Φ-Convex Functions. Differential Geometry-Dynamical Systems, 20, 159-169.
  • Syau, Y.R. and Lee, E.S., 2005. Some Properties of E-Convex Functions. Appl. Math. Lett., 18, 1074–1080.
  • Viloria, J.M. and Cortez, M.V., 2018. Hermite-Hadamard type inequalities for harmonically convex functions on n-coordinates. Appl. Math. Inf. Sci. Lett., 6, 1-6.
  • Yalçın, F.B., 2019. On Some Generalized Inequalities for Two-Dimensional 𝜑-Convex Functions, Journal of Contemporary Applied Mathematics, 9(1), 88-102.
  • Youness, E.A., 1999. E-Convex Sets, E-Convex Functions and E-Convex Programming. J. Optimiz. Theory.App., 102, 439–450.
There are 23 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Fatma Buğlem Yalçın 0000-0003-4276-1820

Nurgül Okur 0000-0002-2544-7752

Publication Date January 15, 2020
Submission Date October 14, 2019
Acceptance Date December 6, 2019
Published in Issue Year 2020

Cite

APA Yalçın, F. B., & Okur, N. (2020). General Convexity of Multidimensional Functions and Related Hermite-Hadamard Type Integral Inequalities. Gümüşhane Üniversitesi Fen Bilimleri Dergisi, 10(1), 241-250. https://doi.org/10.17714/gumusfenbil.632639