On the Occasion of his 80th Birthday Schur Geometric Convexity of Related Function for Holders Inequality with application

In this paper, we investigated the Schur geometric convexity of related function for Holders Inequality by using majorization inequality theory, giving a complete critical condition of Schur Geometrically convex function for Holders Inequality related function and some applications are established. 2020 Mathematics Subject Classifications: 26E60, 26D15, 26A51, 34K38


Introduction
Throughout this paper, we assume that the set of n-dimensional row vector on the real number field by R n . Let (2) Here r l ≥ 0, s l ≥ 0, u > 1, 1 u + 1 v = 1.
The Schur convexity of functions relating to special means is a very significant research subject and has attracted the interest of many mathematicians.There are numerous articles written on this topic in recent years; (see [3] , [6]) and the references therein. As supplements to the Schur convexity of functions, the Schur geometrically convex functions and Schur harmonically convex functions were investigated by Zhang and Yang ([15], [13]), Chu, Zhang and Wang [14], Shi and Zhang ([8], [7]) , Meng, Chu and Tang [4] , Zheng, Zhang and Zhang [17]. These properties of functions have been found to be useful in discovering and proving the inequalities for special means (see [1] -[2], [11], [12]).
Dong-Sheng Wang, Chun -Ru Fu and Huan-Nan Sh [10] investigated the Schur convexity about related function of Holders inequality by using majorization inequality theory . This result gives a full essential condition of Schur convexity for Holders inequality related function, reached sharpen type of Holders inequality Under certain conditions and new inequalities for Stolarsky mean estabilsihed. This paper motivates us to investigate Schur geometric convexity about related function of Holders inequality by using majorization inequality theory.

Preliminaries
To estabilish our main results, we need the following definitions and lemmas.
The function ω : Ψ →R is declining if and just if −ω is escalating.

Main Results
In this paper, by using the principle of majorization as an example, combined with majorization inequality, the Schur-geometrically convexity of Related Function for Holder's Inequality gives sharpening inequality of the Holders under certain conditions. Our primary outcome is as follows: Theorem 1. Let r n ≥ 0 and s n ≥ 0 be any two progressions and let u and v be two non-zero arbitrary real numbers. Let If u ≥ 1, then H 1 (r) is Schur-geometric convex on R + with r 1 , ..., r n and if u < 1, then H 1 (r) is Schur-geometric concave on R + with r 1 , ..., r n .
Proof. : Here H 1 (r) is obviously symmetric with r = r 1 , ..., r n on R + . Let us assume r 1 > r 2 . Now by differentiating (11) partially with respect to r 1 and r 2 , we get It is easy to see that, when u ≥ 1, then 1 ≥ 0 and when u ≤ 1, then 1 ≤ 0.
Theorem 2. Let r n ≥ 0 and s n ≥ 0 be any two progressions and let u and v be two non-zero arbitrary real numbers. Let Proof. : Here H 2 (r) is obviously symmetric with s = s 1 , ..., s n on R + . Let us assume s 1 > s 2 . Now by differentiating (12) partially with respect to s 1 and s 2 , we get Consider, It is easy to see that,when v ≥ 1, then 2 ≥ 0 and when v ≤ 1, then 2 ≤ 0.
Theorem 3. Let φ(x) and ψ(x) be two continuous functions with φ(x) > 0, ψ(x) > 0 and let where u and v are arbitrary real numbers. Let Then H 3 r, s is Schur-geometric concave(convex) with r, s if and only if: Proof. : Here H 3 r, s is obviously symmetric with r = r 1 , r 2 , ..., r n and s = s 1 , s 2 , ..., s n on R + . Let us assume s > r. From (13), we have Now by differentiating this partially with respect to s and r, we get This implies that, have the same symbol.
Hence, we have H 3 r, s is Schur-Geometric concave (convex) with r, s, if and only if: This completes proof of Theorem 3.
Corollary 2. Let φ(x) and ψ(x) be two continuous functions and let their second order derivatives exists with If u, v < 0 and φ(x), ψ(x) are concave functions of opposite monotonicity then H 3 (r, s) is Schur-geometric concave with r = r 1 , r 2 , ..., r n , and s = s 1 , s 2 , ...s n on R + .

Application
The following applications are established by using our main results.
Theorem 5. Let r n ≥ 0 and s n ≥ 0 be any two progressions and let u and v be two non-zero arbitrary real numbers . Then The proof of Theorem 5 is complete.

Conclusion
In this paper, by using of majorization inequality theory we investigated the Schur geometrically convex about related functions of Holders Inequality, giving a complete critical condition of Schur geometrically convex function to Holders Inequality and some applications were established. Despite of these results, the authors are also interested to investigate the results of Schur harmonically convex and m-power convexity about related functions of Holders inequality in future research work.