Geometric Convex Function And Its Generalization

Convex function is a classical concept,closely related to the research of inequality,and plays an important basic and research tool role in mathematics,economics,management,industrial technology and other fields.Geometrically convex function is a parallel concept of convex function,which is given by a transformation or a inequality.It extends the theory and method of control inequality for convex function and is a powerful tool for discovering and proving inequalities.With the introduction of the concept of geometrically convex set,the properties of geometrically convex function based on geometrically convex set have been extensively studied,which further improve the theory of geometrically convexity and enrich the theory of generalized convexity.It has had a huge impact on our daily life through a large number of applications in industry,commerce and medicine.In this paper,by using the methods of literature research,analysis and synthesis,induction and deduction,we further discuss the properties of geometrically convex set,p-geometrically convex set,geometrically convex function and p-geometically convex function,and establish the Hermite-Hadamard type inequalities of m-geometrically convex functions.It is as follows:(1)Firstly,based on the concept of geometrically convex set,the geometrically convex hull and geometrically convex combination of the set are defined,and an equivalent condition to determine the geometrically convex set is given.Then the equialent definition of geometrically convex function is given by using the definition of the epigraph.The invariant property under several operations and the locally global property of geometrically convex function are studied.The judgment theorem of geometrically convex function is also given.(2)Firstly,the concept of geometrically convex set is improved,p-geometrically convex set,p-geometrically convex hull and p-geometrically convex combination are defined,and several equivalent conditions for judging p-geometrically convex set are given.On this basis,the concept of p-geometrically convex function is introduced,and its properties and invariant property under several operations are studied.(3)Some new Hermite-Hadamard type inequalities for m-geometrically convex functions are established,which are based on the integral mean on interval [a,b] of m-geometrically convex functions,the imag of geometric mean on interval [a,b],and the geometric mean of the endpoint of the interval [a,b].