A New Improvement of H\"older inequality via Isotonic Linear Functionals

In this paper, new improvement of celebrated H\"older inequality by means of isotonic linear functionals is established. An important feature of the new inequality obtained in here is that many existing inequalities related to the H\"older inequality can be improved via new improvement of H\"older inequality. We also show this in an application.


Introduction
The famous Young's inequality, as a classical result, state that: if a, b > 0 and t ∈ [0, 1], then with equality if and only if a = b. Let p, q > 1 such that 1/p + 1/q = 1. The inequality (1.1) can be written as (1.2) ab ≤ a p p + b q q for any x, y ≥ 0. In this form, the inequality (1.2) was used to prove the celebrated Hölder inequality. One of the most important inequalities of analysis is Hölder's inequality. It contributes wide area of pure and applied mathematics and plays a key role in resolving many problems in social science and cultural science as well as in natural science.

Hölder's inequality for positive functionals
Let E be a nonempty set and L be a linear class of real valued functions on E having the following properties L1 : If f, g ∈ L then (αf + βg) ∈ L for all α, β ∈ R; where χ E1 is the indicator function of E 1 . It follows from L2 and L3 that χ E1 ∈ L for every E 1 ∈ L.
We also consider positive isotonic linear functionals A : L → R is a functional satisfying the following properties: Isotonic, that is, order-preserving, linear functionals are natural objects in analysis which enjoy a number of convenient properties. Functional versions of well-known inequalities and related results could be found in [1,2,3,4,5,6,8,9].
is an isotonic linear functional. ii , then is an isotonic linear functional.
iii.) If (E, Σ, µ) is a measure space with µ positive measure on E and L = L(µ) then m l=1 a k,l is an isotonic linear functional.
ii iii |f (x, y)| dxdy in the Theorem 5, then the inequality (2.1) reduce the following inequality for double integrals: The aim of this paper is to give a new general improvement of Hölder inequality for isotonic linear functional. As applications, this new inequality will be rewritten for several important particular cases of isotonic linear functionals. Also, we give an application to show that improvement is hold for double integrals.

Main results
Theorem 6. Let L satisfy conditions L1, L2, and A satisfy conditions A1, A2 on a base set E. Let p > 1 and p −1 + q −1 = 1. If α, β, w, f, g ≥ 0 on E and αwf g, βwf g, αwf p , αwg q , βwf p , βwg q , wf g ∈ L then we have i.) Proof. i.) By using of Hölder inequality for isotonic functionals in (2.1) and linearity of A, it is easily seen that ii.) Firstly, we assume that By the inequality (1.1) and linearity of A, we have Finally, suppose that A 1/p (wf p ) A 1/q (wg q ) = 0. Then A 1/p (wf p ) = 0 or A 1/q (wg q ) = 0, i.e. A (wf p ) = 0 or A (wg q ) = 0. We assume that A (wf p ) = 0. Then by using linearity of A we have, Since A (αwf ) , A (βwf ) ≥ 0, we get A (αwf p ) = 0 and A (βwf p ) = 0. From here, it follows that In case of A (wg q ) = 0, the proof is done similarly. This completes the proof.
If we take w = 1 on E in the Theorem 6, then we can give the following corollary: Corollary 1. Let L satisfy conditions L1, L2, and A satisfy conditions A1, A2 on a base set E. Let p > 1 and p −1 + q −1 = 1. If α, β, f, g ≥ 0 on E and αf g, βf g, αf p , αg q , βf p , βg q , f g ∈ L then we have i.) ii.) We can give more general form of the Theorem 6 as follows: Theorem 7. Let L satisfy conditions L1, L2, and A satisfy conditions A1, A2 on a base set E. Let p > 1 and p −1 +q −1 = 1. If α i , w, f, g ≥ 0 on E, α i wf g, α i wf p , α i wg q , wf g ∈ L, i = 1, 2, ..., m, and m i=1 α i = 1, then we have i.) Proof. The proof can be easily done similarly to the proof of Theorem 6.
If we take w = 1 on E in the Theorem 6, then we can give the following corollary: Corollary 2. Let L satisfy conditions L1, L2, and A satisfy conditions A1, A2 on a base set E. Let p > 1 and p −1 +q −1 = 1. If α i , f, g ≥ 0 on E, α i f g, α i f p , α i g q , f g ∈ L, i = 1, 2, ..., m, and m i=1 α i = 1, then we have i.) Corollary 3 (Improvement of Hölder inequality for double integrals). Let p, q > 1 and 1/p + 1/q = 1. If f and g are real |f (x, y)| dxdy in the Corollary 1, then we get the inequality (3.5).

An Application for Double Integrals
In [11], Sarıkaya et al. gave the following lemma for obtain main results.
∂t∂s ∈ L(∆), then the following equality holds: By using this equality and Hölder integral inequality for double integrals, Sarıkaya et al. obtained the following inequality: ∂t∂s q , q > 1, is convex function on the co-ordinates on ∆, then one has the inequalities: If Theorem 8 are resulted again by using the inequality (3.5), then we get the following result: ∂t∂s q , q > 1, is convex function on the co-ordinates on ∆, then one has the inequalities: Proof. Using Lemma 1 and the inequality (3.5), we find Since |f st | q is convex function on the co-ordinates on ∆, we have for all t, s ∈ [0, 1] a combination of (4.3) -(4.5) immediately gives the required inequality (4.2).  . This show us that the inequality (4.2) is better than the inequality (4.1).