tailieunhanh - Convex functions and applications that prove inequality

In this paper, we refer to the concept of function and convex (Convex function) with an application of convex function in solving the problem of proof of Inequality and the Extreme problem about inequality. | Scientific Journal - N 3S 2O19 _5 CONVEX FUNCTIONS AND APPLICATIONS THAT PROVE INEQUALITY Hoang Ngoc Tuyen Hanoi Metropolitaln University Abstract In this paper we refer to the concept of function and convex Convex function with an application of convex function in solving the problem ofproof of Inequality and the Extreme problem about inequality. Key-words Convex function inequalities extreme Email hntuyen@ Received 18 October 2019 Accepted for publication 20 November 2019 1. BACKGROUND Convex analytic plays an important role in the theoretical study of extreme problems. Applied math majors are using convex analytical knowledge and linear space as an effective tool when solving related problems. There are many problems if solved in the usual way is very difficult. However if viewed from a functional perspective the problem solved will be much simpler. In this article please mention such a function class - that is a convex function with the concepts and results involved to apply them to prove some inequalities. 2. SOME PREPARE KNOWLEDGE . Definitions and results Definitions 1 a The function is called convex on paragraph a b if all s A. if and all Ẹ we all have Í . _ - I 1 - i. i fl t _ i - i 1 - if I i Equality occurs when X 6 Ha Noi METROPLOLITAN UNIVERSITy b t A is the concave function on if ừ if f A is the convex function on fự . Definitions 2 The function Aj identified on . - b is called convex if all A 2 T we have 1 Ul x2 n 2 J 2 Equality occurs when 2 Comment If Aj is a continuous function on a b then definition 1 and definition 2 are equivalent. Theorem 1 Suppose the function Aj has a second derivative Aj on b Then 1 If A 0 with all c b then A i concave on paragraph a b 2 If Aj Ũ with all A E n. b than y Aj convex on paragraph a b . Basic properties Properties 1 Give .A J is a convex function continuous on b then every q1 q2 q3 positive such that q1 q2 q3 1 we have Prove We have fJiXi -I- q2x2 Ợ3X3 f . -I- q3 3 i 1 1 2 Qi n Scientific .