MATHEMATICAL S CIENCES AND APPLICATIONS E-NOTES https://doi.org/10.36753/mathenot.685624 10 (2) 63- 71 (2022) - Research Article ISSN: 2147-6268 c MSAEN (m 1 ,m 2 )-Geometric Arithmetically Convex Functions and Related Inequalities Mahir Kadakal Abstract In this manuscript, we introduce and study the concept of (m 1 ,m 2 )-geometric arithmetically (GA) convex functions and their some algebric properties. In addition, we obtain Hermite-Hadamard type inequalities for the newly introduced this type of functions whose derivatives in absolute value are the class of (m 1 ,m 2 )-GA-convex functions by using both well-known power mean and Hölder ’s integral inequalities. Keywords: Convex function; m-convex function; (m1,m2)-GA convex function; Hermite-Hadamard inequality. AMS Subject Classification (2020): 26A51; 26D10; 26D15. 1. Preliminaries and fundamentals Convexity theory provides powerful principles and techniques to study a wide class of problems in both pure and applied mathematics. Hermite-Hadamard integral inequality is very important in the convexity theory. Readers can find more informations in [1–6, 8, 9, 12, 13, 16] and references therein regarding both convexity theory and H-H integral inequalities. Definition 1.1 ([10, 11]). f : I ⊆ R + = (0, ∞) → R is called GA-convex on I if f ( a ξ b 1−ξ ) ≤ ξf (a) + (1 − ξ ) f (b) holds for all a, b ∈ I and ξ ∈ [0, 1]. Definition 1.2 ([14]). f : [0,b] → R is called m-convex for m ∈ (0, 1] if the following inequality f (ξx 1 + m(1 − ξ )x 2 ) ≤ ξf (x 1 )+ m(1 − ξ )f (x 2 ) holds for all x 1 ,x 2 ∈ [0,b] and ξ ∈ [0, 1] . Definition 1.3 ([7]). f : [0,b] → R, b> 0, is caled (m 1 ,m 2 )-convex function, if f (m 1 ξθ + m 2 (1 − ξ )ϑ) ≤ m 1 ξf (θ)+ m 2 (1 − ξ )f (ϑ) for all θ,ϑ ∈ I , ξ ∈ [0, 1] and (m 1 ,m 2 ) ∈ (0, 1] 2 . Received : 06-02-2020, Accepted : 02-03-2022