Generalized trapezoid type inequalities for functions with values in Banach spaces

Document Type : Research Article


Victoria University, Melbourne, Australia.


Let $E$ be a complex Banach space‎. ‎In this paper we show among others that‎, ‎if $\alpha‎ :‎\left[ a,b\right] \rightarrow \mathbb{C}$ is continuous and $Y:‎ ‎\left[ a,b\right] \rightarrow E$ is strongly differentiable on the interval $‎ ‎\left( a,b\right)‎ ,‎$ then for all $u\in \left[ a,b\right]‎ ,‎$‎
‎& \left\Vert \left( \int_{u}^{b}\alpha \left( s\right) ds\right) Y\left(‎
‎b\right)‎ +‎\left( \int_{a}^{u}\alpha \left( s\right) ds\right) Y\left(‎
‎a\right)‎ -‎\int_{a}^{b}\alpha \left( t\right) Y\left( t\right) dt\right\Vert‎
‎& \leq \left\{‎
‎\max \left\{ \int_{u}^{b}\left\vert \alpha \left( s\right) \right\vert‎
‎ds,\int_{a}^{u}\left\vert \alpha \left( s\right) \right\vert ds\right\}‎
‎\int_{a}^{b}\left\Vert Y^{\prime }\left( t\right) \right\Vert dt‎, ‎\\‎
‎‎‎\left[ \int_{u}^{b}\left( b-t\right) \left\vert \alpha \left( t\right)‎
‎\right\vert dt+\int_{a}^{u}\left( t-a\right) \left\vert \alpha \left(‎
‎t\right) \right\vert dt\right] \sup_{t\in \left[ a,b\right] }\left\Vert‎
‎Y^{\prime }\left( t\right) \right\Vert‎ , ‎\\‎ ‎‎
‎\leq \left( b-a\right) ^{1/p}\left[ \left( \int_{u}^{b}\left\vert \alpha‎
‎\left( s\right) \right\vert ds\right) ^{p}+\left( \int_{a}^{u}\left\vert‎
‎\alpha \left( s\right) \right\vert ds\right) ^{p}\right] ^{1/p} \\‎
‎\times \left( \int_{a}^{b}\left\Vert Y^{\prime }\left( t\right) \right\Vert‎
‎^{q}dt\right) ^{1/q}‎‎‎
‎for $p,$ $q>1$ with $\frac{1}{p}+\frac{1}{q}=1.$ Applications for operator‎ ‎monotone functions with examples for power and logarithmic functions are‎ ‎also given‎.


Main Subjects

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