On some inequalities for submultiplicative functions
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ORIGINAL RESEARCH PAPER
On some inequalities for submultiplicative functions Muhammad Aamir Ali1 Zhiyue Zhang1
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Mehmet Zeki Sarikaya2 • Hu¨seyin Budak2
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Received: 9 June 2020 / Accepted: 28 October 2020 Ó Forum D’Analystes, Chennai 2020
Abstract In this work, authors establish Hermite–Hadamard inequalities for submultiplicative functions and give some more inequalities related to Hermite–Hadamard inequalities. We also give new inequalities of Hermite–Hadamard type in the special cases of our main results. Keywords Submultiplicative functions Convex functions Geometrically convex functions and Hermite–Hadamard inequalities
Mathematics Subject Classification 26A09 26D10 26D15 33E20
1 Introduction Submultiplicative functions are of interest because they may be used to parametrize interpolation functors which produce an interpolation Banach algebra when applied to a compatible pair of Banach algebras; (see [1, 2]). Multiplicative and submultiplicative are important concepts both in measure theory and in several fields of mathematics and mathematical inequalities. Especially, submultiplicative & Muhammad Aamir Ali [email protected] Mehmet Zeki Sarikaya [email protected] Hu¨seyin Budak [email protected] Zhiyue Zhang [email protected] 1
Jiangsu Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
2
Department of Mathematics, Faculty of Science and Arts, Du¨zce University, Du¨zce, Turkey
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functions appear naturally in diverse subjects such as interpolation theory [1, 3], and play a significant role in the theory of operators in Orlicz spaces [4]. Inequalities and especially submultiplicative functions theory is one of the most extensively developing fields not only in theoretical and applied mathematics but also physics and other applied sciences. It is well known that submultiplicative functions and inequalities are an important aid in solving diverse problems in mathematics and physics. The paper consists of four sections including introduction, the table of contents gives the general idea of of the paper. In Sect. 2, we establish inequalities of Hermite–Hadamard type for the submultiplicative and multiplicative-starshaped functions. We also present some more inequalities related to Hermite–Hadamard inequalities. In Sect. 3, we give the examples or we can say that we justify our main results with the help of examples. In the final section, we give concluding remarks of our present paper. Definition 1 [5] A function f defined on a set H of real numbers and with range contained in the set Rþ of all positive real numbers, is submultiplicative on H if, for all elements x and y of H such that xy is an element of H fðxyÞ fðxÞfðyÞ: If equality holds, f is called multiplicative; if the inequality is reversed, f is supermultiplicative. A function f is convex on the (possibly infinite) interval D if, for all x and y in D and all t which satisfy 0 t 1,There are a fe
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