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The Hermite-Hadamard type inequality and its estimations via generalized convex functions of Raina type

Year 2021, Volume: 1 Issue: 1, 32 - 43, 30.09.2021
https://doi.org/10.53391/mmnsa.2021.01.004

Abstract

The theory of convexity plays an important role in various branches of science and engineering. The main objective of this work is to introduce the idea of a generalized convex function by unifying s-type m-convex function and Raina type function. In addition, some beautiful algebraic properties and examples are discussed. Applying this new definition, we explore a new sort of Hermite-Hadamard inequality. Furthermore, to enhance the paper we investigate several new estimations of Hermite-Hadamard type inequality. The concepts and tools of this paper may invigorate and revitalize for additional research in this mesmerizing and absorbing field of mathematics.

References

  • Hardy, G.H., Little, J.E. and Polya, G. Inequalities. Cambridge, UK. Cambridge University Press. Cambridge mathematical library, (1952).
  • Kadakal, H. Hermite-Hadamard type inequalities for trigonometrically convex functions. Scientific Studies and Research. Series Mathematic and İnformatics, 28(2), 19-28, (2018).
  • Niculescu, C.P. and Persson, L.E. Convex functions and their applications.Springer, New York, (2006).
  • Tariq, M. New Hermite-Hadamard type inequalities via p-harmonic exponential type convexity and applications. Universal Journal of Mathematics and Applications, 4(2), 59-69, (2021).
  • Özcan, S. and İşcan, İ. Some new Hermite-Hadamard type integral inequalities for the s-convex functions and theirs applications. Journal of Inequalities and Applications, 2019(201), 1-14, (2019).
  • Tariq, M., Nasir, J., Sahoo, S.K. and Mallah, A.A. A note on some Ostrowski type inequalities via generalized exponentially convex function. Journal of Mathematical Analysis and Modeling, 2(2), 1-15, (2021).
  • Khan, M.A., Chu, Y.M., Khan, T.U. and Khan, J. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications. Open Math, 15(1), 1414-1430, (2017).
  • Butt, S.I., Tariq, M., Aslam, A., Ahmad, H. and Nofel, T.H. Hermite-Hadamard type inequalities via generalized harmonic exponential convexity. Journal of Function Spaces, 1-12, (2021).
  • Xi, B.Y. and Qi, F. Some integral inequalities of Hermite-Hadamard type for convex functions with applications to means. Journal of Function Spaces, 1-14, (2012).
  • Butt, S.I., Kashuri, A., Tariq, M., Nasir, J., Aslam, A. and Geo, W. Hermite-Hadamard-type inequalities via n-polynomial exponential-type convexity and their applications. Advances in Difference Equations, 508, (2020).
  • Butt, S.I., Kashuri, A., Tariq, M., Nasir, J., Aslam, A. and Geo, W. n-polynomial exponential-type p-convex function with some related inequalities and their application. Heliyon, 6(11), (2020).
  • Hadamard, J. Étude sur les propriétés des fonctions entiéres en particulier d’une fonction considéréé par Riemann. Journal of Mathematics. Pures. Appl, 58, 171-215, (1893).
  • Raina, R.K. On generalized Wright’s hypergeometric functions and fractional calculus operators. East Asian Mathematical Journal, 21(2), 191-203, (2005).
  • Cortez, M.J.V., Liko, R., Kashuri, A. and Hernández, J.E.H. New quantum estimates of trapezium-type inequalities for generalized φ- convex functions. Mathematics, 7(11), 1047-1066, (2019).
  • Cortez, M.J.V., Kashuri, A. and Hernández, J.E.H. Trapezium-type inequalities for Raina’s fractional integrals operator using generalized convex functions.Symmetry, 12(6), 1034, (2020).
  • Rashid, S., İşcan, İ., Baleanu, D. and Chu, Y.M. Generation of new fractional inequalities via n-polynomials s-type convexity with applications. Advances in Difference Equations, Article ID 264.
  • Barani, A., Ghazanfari, A.G. and Dragomir, S.S. Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex. Journal of Inequalities and Applications, 2012(1), 1-9, (2012).
  • Toplu, T., Kadakal, M. and İşcan, İ. On n-polynomial convexity and some related inequalities. AIMS Math, 5(2), 1304-1318, (2020).
Year 2021, Volume: 1 Issue: 1, 32 - 43, 30.09.2021
https://doi.org/10.53391/mmnsa.2021.01.004

Abstract

References

  • Hardy, G.H., Little, J.E. and Polya, G. Inequalities. Cambridge, UK. Cambridge University Press. Cambridge mathematical library, (1952).
  • Kadakal, H. Hermite-Hadamard type inequalities for trigonometrically convex functions. Scientific Studies and Research. Series Mathematic and İnformatics, 28(2), 19-28, (2018).
  • Niculescu, C.P. and Persson, L.E. Convex functions and their applications.Springer, New York, (2006).
  • Tariq, M. New Hermite-Hadamard type inequalities via p-harmonic exponential type convexity and applications. Universal Journal of Mathematics and Applications, 4(2), 59-69, (2021).
  • Özcan, S. and İşcan, İ. Some new Hermite-Hadamard type integral inequalities for the s-convex functions and theirs applications. Journal of Inequalities and Applications, 2019(201), 1-14, (2019).
  • Tariq, M., Nasir, J., Sahoo, S.K. and Mallah, A.A. A note on some Ostrowski type inequalities via generalized exponentially convex function. Journal of Mathematical Analysis and Modeling, 2(2), 1-15, (2021).
  • Khan, M.A., Chu, Y.M., Khan, T.U. and Khan, J. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications. Open Math, 15(1), 1414-1430, (2017).
  • Butt, S.I., Tariq, M., Aslam, A., Ahmad, H. and Nofel, T.H. Hermite-Hadamard type inequalities via generalized harmonic exponential convexity. Journal of Function Spaces, 1-12, (2021).
  • Xi, B.Y. and Qi, F. Some integral inequalities of Hermite-Hadamard type for convex functions with applications to means. Journal of Function Spaces, 1-14, (2012).
  • Butt, S.I., Kashuri, A., Tariq, M., Nasir, J., Aslam, A. and Geo, W. Hermite-Hadamard-type inequalities via n-polynomial exponential-type convexity and their applications. Advances in Difference Equations, 508, (2020).
  • Butt, S.I., Kashuri, A., Tariq, M., Nasir, J., Aslam, A. and Geo, W. n-polynomial exponential-type p-convex function with some related inequalities and their application. Heliyon, 6(11), (2020).
  • Hadamard, J. Étude sur les propriétés des fonctions entiéres en particulier d’une fonction considéréé par Riemann. Journal of Mathematics. Pures. Appl, 58, 171-215, (1893).
  • Raina, R.K. On generalized Wright’s hypergeometric functions and fractional calculus operators. East Asian Mathematical Journal, 21(2), 191-203, (2005).
  • Cortez, M.J.V., Liko, R., Kashuri, A. and Hernández, J.E.H. New quantum estimates of trapezium-type inequalities for generalized φ- convex functions. Mathematics, 7(11), 1047-1066, (2019).
  • Cortez, M.J.V., Kashuri, A. and Hernández, J.E.H. Trapezium-type inequalities for Raina’s fractional integrals operator using generalized convex functions.Symmetry, 12(6), 1034, (2020).
  • Rashid, S., İşcan, İ., Baleanu, D. and Chu, Y.M. Generation of new fractional inequalities via n-polynomials s-type convexity with applications. Advances in Difference Equations, Article ID 264.
  • Barani, A., Ghazanfari, A.G. and Dragomir, S.S. Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex. Journal of Inequalities and Applications, 2012(1), 1-9, (2012).
  • Toplu, T., Kadakal, M. and İşcan, İ. On n-polynomial convexity and some related inequalities. AIMS Math, 5(2), 1304-1318, (2020).
There are 18 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Research Articles
Authors

Muhammad Tarıq This is me 0000-0001-8372-2532

Hijaz Ahmad This is me 0000-0002-5438-5407

Soubhagya Kumar Sahoo This is me 0000-0003-4524-1951

Publication Date September 30, 2021
Submission Date August 20, 2021
Published in Issue Year 2021 Volume: 1 Issue: 1

Cite

APA Tarıq, M., Ahmad, H., & Sahoo, S. K. (2021). The Hermite-Hadamard type inequality and its estimations via generalized convex functions of Raina type. Mathematical Modelling and Numerical Simulation With Applications, 1(1), 32-43. https://doi.org/10.53391/mmnsa.2021.01.004

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