فهرست نشریات علمی دسترسی آزاد ایران

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 $u\in \left[ a,b\right]‎ ,‎$‎ \begin{align*}‎ ‎& \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\{‎ ‎\begin{array}{l}‎ ‎\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}‎‎‎ ‎\end{array}‎ \right. ‎\end{align*}‎ ‎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.