The application of convex function and GA-convex function

. A convex function is a function that maps from a convex subset of a vector space to the set of real numbers. Convex functions have some important properties, such as non-negativity, monotonicity, and convexity, which can help us derive and prove inequalities. This paper explores the concepts of convex functions and GA-convex functions, demonstrating their utility in proving a variety of common and complex inequalities. Beginning with an overview of convex functions and their extension to GA-convex functions, the study shows how these mathematical tools can be effectively utilized in the context of inequality proofs. By leveraging the properties of these functions, the paper successfully establishes rigorous proofs for a range of inequalities, highlighting the versatility and applicability of convex and GA-convex functions in mathematical analysis. The properties convex and GA-convex functions allow us to use it to determine the direction of inequalities, prove inequalities, determine the optimal solution of inequalities, and even prove Cauchy inequalities.


Introduction
The concavity and convexity of functions have many applications in proving inequalities.Cha conducted research on formulas related to the theorems of convex functions, deriving several important inequalities, which were further applied to prove inequalities and solved conditional extremum problems in 2004 [1].In 2005, Xia derived the Jensen's inequality from the concavity, convexity, and continuity of functions [2].Wu provided the definition of square-convex functions and methods for determining square-convex functions.Then the Jensen-type inequality for square-convex functions was established in 2005 [3].In 2010, Song and Wan obtained a more concise Hadamard-type inequality for GA-convex functions through their study of GA-convex functions [4].Shi et al. obtained a new refinement of the Hermite-Hadamard-type inequality for GA-convex functions in 2013 [5].In the same year, Shi et al. derived some new weighted Hadamard-type inequalities for differentiable GA-convex functions [6].Wu and Mao proved the Hermite-Hadamard inequality on a special region in 2022 [7].
This article mainly introduces convex functions and GA-convex functions.The paper first introduces the definition of convex functions and its equivalent definitions, extends it to n numbers, and then proves several common inequalities using its properties in section 2. This paper transitions from convex functions to GA-convex functions, introduces its definition, proves its properties, creates an inequality, and then proves a more complex inequality relationship in section 3.

Properties of concave-convex function
The definition of concave-convex function will be introduced first, followed by an explanation of its properties.
Proof: Let If the  1 and  2 in equations ( 1) and ( 3) are interchanged, the result remains unchanged.This means that the above results are independent of whether  1 is greater than or less than  2 , as long as  ∈ ( 1 ,  2 ).Therefore, set So () is concave upwards or downwards in interval [, ] that can be replaced by another form: Then it indicates that () is concave up or concave down on the interval [, ].

The Application of Convex Functions in Proving Inequalities
In this subsection, common inequalities are proven using the properties of convex functions.First, a lemma is introduced.Lemma 2.4.Each Let () be convex upwards and downwards on [, ], ∀ 1 ,  2 , … ,   ∈ [, ], there exists, Proof: By induction, when  = 1,2, the proposition can be proven using (6).Assuming it holds for  =  , prove that it also holds for Example 2.5.Let  1 ,  2 , … ,   > 0. Prove: Proof: First prove the right half of the equation.
The inequality can be proven using convex function () =  and the Lemma 2.4.Replacing   with 1   ( = 1,2, … , ) can prove the left half of the inequality.

Characteristics of GA-Convex Functions
The definition of GA-Convex Functions will be introduced first, followed by an explanation of its properties.
Proof: It is easy to establish the connection between the second derivative of (  ) on the interval (, ) and the concavity/convexity of the function., it is easy to infer  ∈ (0,1).By the properties of GA-Concave, the following formula can be derived.
Dividing both sides by  −  will get the inequality on the right-hand side.By the same way, the inequality on the left-hand side can be proved.Let △=  − ,  +   △∈ [, ],  = 1,2, … , .By the definition of a definite integral and Theorem 3.4, the following formula can be derived.
Replacing  and b with √ and √ in (21) results in  2  4 can be obtained.Only the last inequality needs to be proven now.
Therefore, the inequality Example 3.6 is proved.

Conclusion
This article first introduces the definition of convex functions from a geometrically intuitive perspective, then extends from two points on an interval to n points, skillfully demonstrating that the harmonic mean is less than or equal to the geometric mean, which is less than or equal to the arithmetic mean.In the subsequent section, it extends the ordinary convex functions to GA-convex functions, studies their sufficient and necessary conditions and properties, and ultimately constructs an inequality to prove the complex inequality chain in the example.It is evident that convex functions can easily be used to prove seemingly complex inequalities, but they also require assistance from other tools in mathematical analysis.It is hoped that in the future, building upon the foundation laid by this research, researchers can continue to advance the understanding and application of convex functions in the realm of inequalities.