Schur m-Power Convexity of a Class of Multiplicatively Convex Functions and Applications

and Applied Analysis 3 Recently, Yang presents the Schur f-convexity in [7] as follows. Definition 8 (see [7–9]). Let Ω ⊆ Rn be a set with nonempty interior andf a strictly monotone function defined onΩ. Let f (x) = (f (x 1 ) , f (x 2 ) , . . . , f (x n )) , f (y) = (f (y 1 ) , f (y 2 ) , . . . , f (y n )) . (12) Then function φ : Ω → R is said to be Schur f -convex onΩ if f(x) ≺ f(y) onΩ implies ψ(x) ≤ ψ(y). ψ is said to be Schur f -concave if −ψ is Schur f -convex. Take f(x) = x, lnx, x−1 in Definition 8, it yields the Schur convexity, Schur geometrical convexity, and Schur harmonic convexity. It is clear that the Schur f -convexity is a generalization of the Schur convexity mentioned above. In general, we have the following. Definition 9 (see [7–9]). Let f : R ++ → R be defined by f(x) = (x m − 1)/m if m ̸ = 0 and f(x) = lnx if m = 0. Then function ψ : Ω ⊆ Rn ++ → R is said to be Schur m-power convex on Ω if f(x) ≺ f(y) on Ω implies ψ(x) ≤ ψ(y). ψ is said to be Schur m-power concave if −ψ is Schur mpower convex. Lemma 10 (see [7–9]). Let ψ : Ω ⊆ Rn ++ → R be continuous onΩ and differentiable inΩ. Thenψ is schurm-power convex (Schurm-power concave) onΩ if and only if ψ is symmetric on Ω and

Recently, Yang [7][8][9] generalized the notion of Schur convexity to Schur m-power convexity, which contains the Schur convexity, Schur geometrical convexity, and Schur harmonic convexity.Moreover, he discussed Schur m-power convexity of Stolarsky means [7], Gini means [8], and Daróczy means [9].Wang and Yang showed that generalized Hamy symmetric function [10] and a class of symmetric functions [11] are Schur m-power convex.Now we define the more general dual form of symmetric function.
Definition 1.Let Ω ⊂ R  ++ be a symmetric convex set with nonempty interior and  : Ω → R ++ is continuous on Ω and differentiable in the interior of Ω.For  > 0, define the symmetric functions  , (x, ) by  , (x, ) = ∏ In this paper, we investigate the Schur m-power convexity of the above more general dual form of symmetric functions.In particular, we obtain that the above more general dual form of symmetric functions is Schur geometrically convex and Schur harmonically convex, which generalizes some known results.As a consequence, we are able to prove a number of new inequalities concerning the th power mean, the arithmetic mean, and the geometric and the harmonic mean.

Definitions and Lemmas
We first recall several definitions as follows.
(2) Let Ω ⊂ R  ,  : Ω → R is said to be increasing if x ≥ y implies (x) ≥ (y). is said to be decreasing if and only if − is increasing.
(1) y majorizes x (in symbols are rearrangements of x and y in a descending order. (2) A real-valued function is a Schur concave function on Ω if and only if − is a Schur convex function.
The following Theorem is basic and plays an important role in the theory of the Schur geometrically convex function.
( Lemma 7 (see [15,16]).Let Ω ∈ R  ++ be a symmetric and harmonically convex set with inner points and let  : Ω → R ++ be a continuously symmetric function which is differentiable in Ω 0 .Then  is Schur harmonically convex (Schur harmonically concave) on Ω if and only if Schur convex, Schur geometrically convex, and Schur harmonically convex were introduced by Marshall et al. [13], Zhang [14], and Chu and Sun [15], respectively, and played a key role in analytic inequalities .Moreover, the theory of convex functions and Schur convex functions is one of the most important research fields in modern analysis and geometry.
Then function  : Take () = , ln ,  −1 in Definition 8, it yields the Schur convexity, Schur geometrical convexity, and Schur harmonic convexity.It is clear that the Schur f -convexity is a generalization of the Schur convexity mentioned above.In general, we have the following.Definition 9 (see [7][8][9]).Let  : R ++ → R be defined by is said to be Schur m-power concave if − is Schur mpower convex.
The following lemma is clearly due to the monotonicity property of the function   on R ++ .
(i) f is multiplicatively convex.
Moreover, if  is twice differentiable, then  is multiplicatively convex if and only if

Main Results and Proof
Our main results are stated as follows.
Lemma 20.Let the function  :  ⊂ R ++ → R ++ be continuous on Ω and differentiable in the interior of Ω.For  ≤ 0, if  is increasing and multiplicatively convex, then Proof.Since  is multiplicatively convex, and by using Lemma 15, we can easily see that   ()/() is increasing.Further, by applying () ≥ 0 and the monotonicity of , it follows that  1−   ()/() is also increasing for  > 0 and  ≤ 0.
Proof.We can easily derive that So the function () is increasing.
Because  is multiplicatively convex, and by Lemma 15, we get On the other hand, for  > 0 and  ≤ 0, we easily know that the functions   and (  +    ) 1/ are increasing about .
Proof of Theorem 16.By Lemma 10 and Remark 12, we only need to prove that To prove the above inequality, we consider the following three cases for .
Case 2. For  = 2,  > 0, we have We can easily derive that By differentiating the above equation with respect to  1 , we obtain Similarly, we have So, from ( 34) and (35), and by applying Lemma 22, we have So we get that Δ 2 ≥ 0.
Case 3.For 3 ≤  ≤ ,  > 0, similarly to the discussion of Case 2, we have By differentiating to the above with respect to  1 , we have Similarly, we can have From ( 38) and ( 39), we have So we get that Δ 2 ≥ 0. So the proof of Theorem 16 is complete.

Applications
Let   > 0,  = 1, 2, . . .,  be  positive real numbers and set x = { 1 ,  2 , . . .,   }.The th power mean of of order  ∈ R of   is defined by In particular, for  = 1,  = 0, and  = 1 we, respectively, get the arithmetic, the geometric, and the harmonic means of   , and set In this section, some applications of the results in Section 3 are given.Some analytic inequalities are established.In particular, several inequalities involving the th power mean and the arithmetic, the geometric, or the harmonic means are presented.

Analytic Inequalities.
To establish some analytic inequalities, we first give a lemma.
By using Theorem 25 and Lemma 24, we get the following inequalities.
By using Theorem 33 and Lemma 24, we get the following inequalities.
By using Theorem 35 and Lemma 24, we get the following inequalities.

Geometric Inequalities.
In this section, some geometric inequalities of -dimensional simplex are established by use of the results of Theorem 16.Lots of geometric inequalities for an -dimensional simplex are established (see [37][38][39][40][41][42]). In this section, applying the above Lemma and the main results in Section 2, we establish some interesting geometric inequalities on -dimensional simplex in -dimensional Euclidean space   .
In what follows, Let Ω = { 1 ,  2 , . . .,  +1 } be an -dimensional simplex in -dimensional Euclidean space   ( ≥ 2) with  the volume.We denote by ℎ  ,   ,   ( = 1, 2, . . .,  + 1), and  the altitudes, the radii of excircles, the areas of lateral surfaces, and the inradius of Ω, respectively.For a given point  in Ω, let   stand for the intersection point of straight line    and hyperplane We first give some lemmas.