Abstract

In the paper, by using the properties of Schur m-power convex function, we discuss Schur m-power convexity of a new class of symmetric functions where i₁, i₂, …, i_r are non-negative integers, and p ∈ N⁺. We obtain that is Schur m-power convex for m ≤ 0 and Schur m-power concave for m ≥ p. We also give a counter example to illustrate is neither Schur convex nor Schur concave for p>1. As applications, a Klamkin-Newman type inequality and some analytic inequalities are derived.

1. Introduction

Throughout this paper, let

and

In 1923, Schur had firstly introduced the concept of Schur convexity ( ¹). In 1970, Schur convexity was generalized and the notion of Schur harmonic convexity was brought in ². In 2004, Zhang had defined the Schur geometrical convexity as a parallel one to Schur convex theory ( ³). In 2012, Yang generalized the notion of Schur convexity to Schur f-convexity, which contains the Schur convexity, the Schur geometrical convexity, Schur harmonic convexity and so on ( ^{4, 5, 6}). Further, Schur m-power convexity of some special means have been discussed in ^{4, 5, 6}. The Schur convex theory have played an important role in the investigation of Mathematics and other disciplines. In recent years, the study on the properties of the symmetric functions and the properties of the means is very active by using theory of majorization and Schur convexity. Subsequently, lots of new analytic inequalities were obtained and many classical inequalities were generalized and improved. Related work please see References [2,3,6,7-32].

In ¹⁵, the Hamy symmetric function was defined as following:

where are non-negative integers, , and . Its properties and applications can be found in ⁸.

In ¹¹, Guan generalized the Hamy symmetric function and defined the generalized Hamy symmetric function:

where are non-negative integers, , and .

Guan also proved that Hamy symmetric function and generalized Hamy symmetric function are Schur concave and Schur geometrically convex in , and established some analytic inequalities by use of the theory of majorization ¹¹.

In ⁹, Chu and Sun proved the generalized Hamy symmetric function is Schur harmonically convex in and established some analytic inequalities as its applications.

In ²⁴, the following symmetric function was defined by Wang:

whereandare non-negative integers. Wang proved that is Schur geometrically convex and Schur m-power convex for m≤0 and listed a counter example to illustrate is neither Schur convex nor Schur concave.

Now, we define the following new symmetric function:

Definition 1.1. Let . For fixed , define the symmetric function as following

(1.1)

where are non-negative integers.

Obviously, is a factor of .

In the paper, by using the properties of Schur m-power convex function, we discuss Schur m-power convexity of a new class of symmetric functions We obtain that is Schur m-power convex for and Schur m-power concave for . We also give a counter example to illustrate is neither Schur convex nor Schur concave for As applications, a Klamkin-Newman type inequality and some analytic inequalities are derived.

2. Preliminary

Firstly, we recall some necessary definitions and lemmas.

Definition 2.1 ( ¹). Assume that and are two n-tuples real numbers.

(1) y majorizes x (in symbols ), if

and

where are rearrangements of x and y in a descending order.

(2) Let , then is said to be a convex set if

for any and , where

A function is said to be Schur convex on if

A function is said to be Schur concave on if and only if −f is a Schur convex function.

Lemma 2.1 ( ¹). Let be a convex set of and has a nonempty interior set. Assume that is a symmetric function which is continuous on and differentiable in . Then f is a Schur convex function (or a Schur concave function) if and only if

(2.1)

Definition 2.2. Let be a subset of ,

(1) is said to be a harmonically convex set if

where

Let be a symmetric function and has continuous partial derivatives on . Then f is a Schur harmonically convex function if

where . f is called Schur harmonically concave if −f is Schur harmonically convex.

Lemma 2.2 ( ²). Let be a symmetric harmonically convex set with a nonempty interior. Assume that is a symmetric function which is continuous on and differentiable in. Then f is a Schur harmonically convex function (or a Schur harmonically concave function) if and only if

(2.2)

Definition 2.3 ( ³). Let be a subset of ,

(1) The n-tuple x is said to be geometrically majorized by y (in symbols ), if

and

(2) is said to be a geometrically convex set if

where

(2) A function is Schur geometrically convex function if

A function is called Schur geometrically concave if −f is Schur geometrically convex.

Lemma 2.3 ( ³). Let be a symmetric geometrically convex set with a nonempty interior. Assume that is a symmetric function which is continuous on and differentiable in. Then f is a Schur geometrically convex function (or a Schur geometrically concave function) if and only if

(2.3)

Definition 2.4 ( ^{4, 5, 6}). Let be defined by

(2.4)

Then function is said to be Schur m-power convex on if

(2.5)

A function is said to be Schur m-power concave if is Schur m-power convex.

If taking in Definition 2.4, then the concepts of Schur-convex, Schur-geometrically convex and Schur-harmonically convex functions can be deduced respectively.

Lemma 2.4 ( ^{4, 5, 6}). Let be a symmetric set with a nonempty interior . Assume that is a symmetric function which is continuous on and differentiable in. Then is a Schur m-power convex (m-power concave) on if and only if

(2.6)

hold for any with

The relation of different orders of Schur m-power convex function have been studied by Zhang in ³⁰ and the following result were obtained.

Lemma 2.5 ( ^{22, 30}). Let, and a real function . If f is a increasing and Schur p-power convex function, then f must be Schur q-power convex function.

Lemma 2.6 ( ¹). Let be open and convex set, and a real function is differentiable on Ω. Then is increasing if and only if , where

3. Main Results

In this section, the Schur m-power convexity of for and is discussed.

Theorem 3.1. Let For fixed

(3.1)

where are non-negative integers.

(1) when is Schur m-power convex.

(2) when is Schur m-power concave.

Proof. Obviously, the function is symmetric and continuously differentiable.

By calculation, it follows that

(3.2)

Make

(3.3)

By calculating the partial derivatives of L, Q, R and S on and , respectively, it is easy to obtain

(3.4)

Thus by differentiating with respect to , we can derive

(3.5)

Therefore, it deduce that

(3.6)

Because the function is increasing for and decreasing for in , then for

for

By means of Lemma 2.5, the Theorem is proved.

Remark 3.1. When is neither Schur convex nor Schur concave.

In fact, if is Schur convex for and then is also Schur convex.

For ,

Thus

which contradicts with .

If is Schur concave forr and then

is also Schur concave.

That is

which contradicts with for .

Corollary 3.1.1. For and we have

(1) is Schur harmonically convex.

(2) is Schur geometrically convex.

Proof. In Theorem 3.2, m takes −1 and 0, respectively, we get the results.

Corollary 3.1.2. For and is Schur concave.

Proof. In Theorem 3.2, m and p take 1, the results are obtained.

4. Applications

The following lemmas are useful for establishing the inequalities.

Lemma 4.1 ( ¹²). Let and . If , then

(4.1)

Lemma 4.2 ( ¹²). Let and . If , then

(4.2)

Let and , then the mean

is famously the -th power mean of order of . Specially, taking, and , respectively, the arithmetic, the geometric and the harmonic means of are derived as following:

Lemma 4.3 ( ²⁸). Assume that , then

(4.3)

(4.4)

(4.5)

By making use of above Lemmas and Corollary 3.1.1 and 3.1.2, the following results are easy to be get.

Theorem 4.1. Assume that , and . If and fixed , then

(4.6)

(4.7)

(4.8)

(4.9)

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

The following inequality

(4.15)

whereand is well known Klamkin-Newman inequality ( ³¹).

Putting and respectively in (4.8), the Klamkin-Newman type inequalities are also derived.

Corollary 4.1.1. For for , then

Specially, when , we have

For in , it follows that

For simplicity, assume that

Since is harmonically convex function on for and , it is easy to derive the following inequality by means of the inequality (4.8).

Corollary 4.1.2. For , then

Further,

5. Conclusion

To sum up, we define a new class of symmetric functions in this paper

where are non-negative integers, , and , and discuss Schur m-power convexity of by using the properties of Schur m-power convex function. We obtain that is Schur m-power convex for and Schur m-power concave for . We also give a counter example to illustrate is neither Schur convex nor Schur concave for . As applications, a Klamkin-Newman type inequality and some analytic inequalities are derived.

Acknowledgements

This work was supported by PhD Research Foundation of Inner Mongolia University for Nationalities (No. BS401 and No. BS402) and Science Research Foundation of Inner Mongolia University for Nationalities (No. NMDYB1777) and Natural Science Foundation of Inner Mongolia Autonomous Region (No. 2019MS01007).

References