An International Journal of Optimization and Control: Theories & Applications ISSN:2146-0957 eISSN:2146-5703 Vol.9, No.2, pp.216-222 (2019) http://doi.org/10.11121/ijocta.01.2019.00738 RESEARCH ARTICLE Some integral inequalities for multiplicatively geometrically P -functions Huriye Kadakal Ministry of Education, Bulancak Bah¸ celievler Anatolian High School, Giresun, Turkey huriyekadakal@hotmail.com ARTICLE INFO ABSTRACT Article History: Received 24 October 2018 Accepted 15 March 2019 Available 31 July 2019 In this manuscript, by using a general identity for differentiable functions we can obtain new estimates on a generalization of Hadamard, Ostrowski and Simpson type inequalities for functions whose derivatives in absolute value at certain power are multiplicatively geometrically P -functions. Some applica- tions to special means of real numbers are also given. Keywords: Multiplicatively P -functions Multiplicatively geometrically P -function AMS Classification 2010: 26A51; 26D15 1. Preliminaries Let function ψ : I ⊆ R → R be a convex defined on an interval I of real numbers and ζ,η ∈ I with ζ<η. The following ψ  ζ + η 2  ≤ 1 η − ζ η  ζ ψ(u)du ≤ ψ(ζ )+ ψ(η) 2 . (1) holds. This double inequality is known in the lit- erature as Hermite-Hadamard integral inequality for convex functions [1]. Both inequalities hold in the reversed direction if the function ψ is concave. Let ψ : I ⊆ R → R be a mapping differentiable in I ◦ , the interior of I , and let ζ,η ∈ I ◦ with ζ<η. If |ψ ′ (x)|≤ M for all x ∈ [ζ,η] , then we hold the following inequality        ψ(x) − 1 η − ζ η  ζ ψ(t)dt        ≤ M η − ζ  (x − ζ ) 2 +(η − x) 2 2  for all x ∈ [ζ,η] . This inequality is known as the Ostrowski inequality [2]. The following inequality is well known as Simp- son’s inequality . Let ψ :[ζ,η] → R be a four-times continuously differentiable mapping on (ζ,η) and   ψ (4)   ∞ = sup x∈(ζ,η)   ψ (4) (x)   < ∞. Then the following inequal- ity        1 3  ψ(ζ )+ ψ(η) 2 +2ψ  ζ + η 2  − 1 η − ζ η  ζ ψ(u)du        ≤ 1 2880   ψ (4)    ∞ (η − ζ ) 4 . holds. Definition 1. A nonnegative function ψ : I ⊆ R → R is called P -function if ψ (tζ + (1 − t) η) ≤ ψ (ζ )+ ψ (η) holds for all ζ,η ∈ I and t ∈ (0, 1). 216