New Refinements for integral and sum forms of H\"older inequality

In this paper, new refinements for integral and sum forms of H\"older inequality are established. We note that many existing inequalities related to the H\"older inequality can be improved via obtained new inequalities in here, we show this in an application.


Introduction
The famous Young inequality for two scalars is the t-weighted arithmetic-geometric mean inequality. This inequality says that if x, y > 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) xy ≤ x p p + y 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.
Of course the Hölder's inequality has been extensively explored and tested to a new situation by a number of scientists. Many generalizations and refinements for Hölder's inequality have been obtained so far. See, for example, [1,3,4,5,6,7,8,9,10,11] and the references therein. In this paper, by using a simple proof method some new refinements for integral and sum forms of Hölder's inequality are obtained.

Main Results
Theorem 3. Let p > 1 and 1/p + 1/q = 1. If f and g are real functions defined Proof. i.)First method for Proof (Short method): By using of Hölder inequality in (1.3), it is easily seen that Second method for Proof (Long method): Applying (1.3) on the subinterval [a, λb + (1 − λ)a] and on the subinterval [λb + (1 − λ)a, b], respectively, we get Adding the resulting inequalities, we get: By the change of variable x = ub + (1 − u)a; on the right hand sides integrals in Integrating both sides of this inequality over [0, 1] with respect to λ we obtain that Then, By applying of Hölder inequality for the right hand sides integrals in the last

By Fubini theorem and the change of variable
Therefore the inequality (2.2) is trivial in this case.
Finally, we consider the case Applying (1.1) on the right hand sides integrals of the last inequality This completes the proof. The more general versions of Theorem 3 can be given as follow: Proof. The proof of Theorem is easily seen by using similar method the proof of Theorem 3.

Remark 2.
It is easily seen that the inequalities obtained in Theorem 4 are the best than the inequality (1.3).
ii.) In the inequality (2.4) of Theorem 4, if we take α(x) = b−x b−a and β(x) = x−a b−a , then we have the inequality (2.1).
The proof of Theorem is easily seen by using similar method the proof of Theorem 3.

Remark 5.
It is easily seen that the inequalities obtained in Theorem 6 are the best than the inequality (1.4).

An Application
In [2], Dragomir et al. gave the following lemma for obtain main results.
, then the following equality holds: By using this equality and Hölder integral inequality Dragomir et al. obtained the following inequality: Theorem 7. Let f : I • ⊆ R → R be a differentiable mapping on I • , a, b ∈ I • with a < b. If the new mapping |f ′ | q is convex on [a, b], then the following inequality holds: If Theorem 7 are resulted again by using the inequality (2.1) in Theorem 3, then we get the following result: , then the following inequality holds: where 1/p + 1/q = 1.

10İMDATİŞCAN
Proof. Using Lemma 1 and the inequality (2.1), we find Using the convexity of |f ′ | q , we have