On the certain abstract convexity classes and their applications

In this study, by using the theorem given for a function which is abstract con- cave with respect to set of quadratic functions, well-known mean inequalities are sharpened under proper conditions. Secondly, at the one step of the Cutting Angle Method which is used to solve the minimum of the increasing positively homoge- neous function over unit simplex, there exists a subproblem to find minimum of the function constructed by minimum type functions. For the solution of this subprob- lem, by considering problem in a geometric way, a new algorithm is suggested and its efficiency is proved teoretically and practically by comparing to old algorithm in some examples. It is seen that the suggested algorithm empowers and accelerates the method in solving of the problem.