Hölder’s type generalizations of Opial’s inequalities for two absolutely continuous functions and applications

Abstract

In this paper we establish some Hölder’s type generalizations of Opial’s inequalities for two absolutely continuous functions. Applications related to the trapezoid weighted inequalities and to Fejér’s inequality for convex functions are also provided. Some Grüss’ type inequalities for p -norms are pointed out. Certain examples for norm inequalities in Hilbert spaces and approximations of Finite Fourier transform are given as well.

Among others we show that, if h ,

are absolutely continuous on

a , b $\left[a,b\right]$

with

h ′ ∈ L p a , b ${h}^{\prime }\in {L}_{p}\left[a,b\right]$

,

k ′ ∈ L q a , b , k a = k b = 0 ${k}^{\prime }\in {L}_{q}\left[a,b\right],k\left(a\right)=k\left(b\right)=0$

and p , q > 1 with

1 p + 1 q = 1 $\frac{1}{p}+\frac{1}{q}=1$

, then

∫ a b h ′ t k t d t ≤ ∫ a b K t h ′ t p d t 1 / p ∫ a b a + b 2 − t k ′ t q d t 1 / q ≤ ∫ a b 1 p K t h ′ t p d t + 1 q a + b 2 − t k ′ t q d t , \begin{align*}\hfill & {\int }_{a}^{b}\left\vert {h}^{\prime }\left(t\right)k\left(t\right)\right\vert \mathrm{d}t\hfill \\ \hfill & \le {\left({\int }_{a}^{b}K\left(t\right){\left\vert {h}^{\prime }\left(t\right)\right\vert }^{p}\mathrm{d}t\right)}^{1/p}{\left({\int }_{a}^{b}\left\vert \frac{a+b}{2}-t\right\vert {\left\vert {k}^{\prime }\left(t\right)\right\vert }^{q}\mathrm{d}t\right)}^{1/q}\hfill \\ \hfill & \le {\int }_{a}^{b}\left[\frac{1}{p}K\left(t\right){\left\vert {h}^{\prime }\left(t\right)\right\vert }^{p}\mathrm{d}t+\frac{1}{q}\left\vert \frac{a+b}{2}-t\right\vert {\left\vert {k}^{\prime }\left(t\right)\right\vert }^{q}\right]\mathrm{d}t,\hfill \end{align*}