On extension of an inequality including Arithmetic and Logarithmic means via generalized Hermite-Hadamard inequality

Subject Areas : Statistics

Mohsen Rostamian Delavar ¹ , Mohsen Kian ²

1 - Department of Mathematics, Faculty of Basic Sciences, University of Bojnord,, Iran
2 - Department of Mathematics, Faculty of Basic Sciences, University of Bojnord,, Iran

Received: 2020-12-04 Accepted : 2023-09-12 Published : 2024-10-17

Keywords: میانگین عددی خاص, نامساوی هرمیت-هادامارد, تابع "M-لیپشیتس", تابع کراندار,

Abstract :

We present an extension version of the well-known Hermite-Hadamard inequality by using the definition of the mapping L(t) and by using the convexity of considered function. This inequality has several applications in mathematical inequalities which in special case gives a generalization for a mean type inequality including Arithmetic mean and generalized Logarithmic mean, which is known in the field of mathematical means with many applications. In fact we extended an Arithmetic-generalized Logarithmic mean type inequality with a natural number as the power to a generalized real number type inequality, by the use of results obtained in this work. Also some new properties related to the mapping L(t) are investigated. Furthermore, at last, some estimation type inequalities for the case that considered function is M-Lipschitz and the case that is bounded are given. In fact by using the definition of the mapping L(t) we obtain an estimation type results for the difference of Arithmetic mean and generalized Logarithmic mean.

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