If $A,B\in \mathcal{B}\left(\mathcal{H}\right)$ are normal accretive operators, $X\in \mathcal{B}\left(\mathcal{H}\right),0<\alpha <1$ and Φ is a s.n. function, we proved that $\begin{array}{rl}& |\left(A^{\ast }+A{)}^{1-\alpha }X\right(B+B^{\ast }{)}^{\alpha }{|}^2\\ & ⩽\mathrm{\Gamma }(2-2\alpha ){\int }_{[0,+\mathrm{\infty })}{\mathrm{e}}^{-t{B}^{\ast }}\left(B^{\ast }+B{)}^{\alpha }\right|AX+XB{|}^2\\ & \times (B^{\ast }+B{)}^{\alpha }{\mathrm{e}}^{-tB}{t}^{2\alpha -1}\mathrm{d}t,\end{array}$ $\begin{array}{rl}& \parallel \left(A^{\ast }+A{)}^{1-\alpha }X\right(B+B^{\ast }{)}^{\alpha }{\parallel }_{\mathrm{\Phi }}⩽\mathrm{\Gamma }(2-2\alpha )\mathrm{\Gamma }\left(2\alpha \right)\parallel AX+XB{\parallel }_{\mathrm{\Phi }},\\ & \mathrm{i}\mathrm{f}AX+XB\in {\mathcal{C}}_{\mathrm{\Phi }}\left(\mathcal{H}\right).\end{array}$Let $A,B,X\in \mathcal{B}\left(\mathcal{H}\right),A⩾0,B⩾0,\eta ,\theta \in \mathbb{R},\alpha \in (0,1)$ and Φ be a s.n. function. If ${\mathrm{e}}^{\mathrm{i}\eta }AX+{\mathrm{e}}^{\mathrm{i}\theta }XB\in {\mathcal{C}}_{\mathrm{\Phi }}\left(\mathcal{H}\right),$ then we have the following generalization of Young's norm inequality in Jocić [Cauchy–Schwarz norm inequalities for weak*-integrals of operator valued functions. J Funct Anal. 2005;218:318–346], Corollary 4.1 $\left|{\mathrm{e}}^{\mathrm{i}\eta }+{\mathrm{e}}^{\mathrm{i}\theta }\right|\parallel A^{1-\alpha }X{B}^{\alpha }{\parallel }_{\mathrm{\Phi }}⩽\sqrt{\mathrm{\Gamma }(2-2\alpha )\mathrm{\Gamma }\left(2\alpha \right)}\parallel {\mathrm{e}}^{\mathrm{i}\eta }AX+{\mathrm{e}}^{\mathrm{i}\theta }XB{\parallel }_{\mathrm{\Phi }}.$Various examples and applications of the obtained norm inequalities are also presented, including those related to the Heinz and Heron means and Zhan inequalities.

如果$\mathrm{一种},乙\in \mathcal{乙}\left(\mathcal{H}\right)$是正常的增生算子，$X\in \mathcal{乙}\left(\mathcal{H}\right),0<\alpha <\mathrm{1个}$并且 Φ 是一个 sn 函数，我们证明了$\begin{array}{rl}& |\left({\mathrm{一种}}^{＊}+\mathrm{一种}{)}^{\mathrm{1个}-\alpha }X\right(乙+{乙}^{＊}{)}^{\alpha }{|}^{\mathrm{2个}}\\ & ⩽\mathrm{\Gamma }(\mathrm{2个}-\mathrm{2个}\alpha ){\int }_{[0,+\mathrm{\infty })}{\mathrm{电子}}^{-吨{乙}^{＊}}\left({乙}^{＊}+乙{)}^{\alpha }\right|\mathrm{一种}X+X乙{|}^{\mathrm{2个}}\\ & \times ({乙}^{＊}+乙{)}^{\alpha }{\mathrm{电子}}^{-吨乙}{吨}^{\mathrm{2个}\alpha -\mathrm{1个}}\mathrm{d}吨,\end{array}$ $\begin{array}{rl}& \parallel \left({\mathrm{一种}}^{＊}+\mathrm{一种}{)}^{\mathrm{1个}-\alpha }X\right(乙+{乙}^{＊}{)}^{\alpha }{\parallel }_{\mathrm{\Phi }}⩽\mathrm{\Gamma }(\mathrm{2个}-\mathrm{2个}\alpha )\mathrm{\Gamma }\left(\mathrm{2个}\alpha \right)\parallel \mathrm{一种}X+X乙{\parallel }_{\mathrm{\Phi }},\\ & \mathrm{一世}\mathrm{F}\mathrm{一种}X+X乙\in {\mathcal{C}}_{\mathrm{\Phi }}\left(\mathcal{H}\right).\end{array}$让$\mathrm{一种},乙,X\in \mathcal{乙}\left(\mathcal{H}\right),\mathrm{一种}⩾0,乙⩾0,\eta ,\theta \in \mathbb{R},\alpha \in (0,\mathrm{1个})$和 Φ 是一个 sn 函数。如果${\mathrm{电子}}^{\mathrm{一世}\eta }\mathrm{一种}X+{\mathrm{电子}}^{\mathrm{一世}\theta }X乙\in {\mathcal{C}}_{\mathrm{\Phi }}\left(\mathcal{H}\right),$然后我们对 Jocić [Cauchy–Schwarz 范数不等式的弱*-算子值函数积分的范数不等式进行了以下推广。J 功能肛门。2005;218:318–346]，推论 4.1$\left|{\mathrm{电子}}^{\mathrm{一世}\eta }+{\mathrm{电子}}^{\mathrm{一世}\theta }\right|\parallel {\mathrm{一种}}^{\mathrm{1个}-\alpha }X{乙}^{\alpha }{\parallel }_{\mathrm{\Phi }}⩽\sqrt{\mathrm{\Gamma }(\mathrm{2个}-\mathrm{2个}\alpha )\mathrm{\Gamma }\left(\mathrm{2个}\alpha \right)}\parallel {\mathrm{电子}}^{\mathrm{一世}\eta }\mathrm{一种}X+{\mathrm{电子}}^{\mathrm{一世}\theta }X乙{\parallel }_{\mathrm{\Phi }}.$还介绍了所获得范数不等式的各种示例和应用，包括与海因茨和海伦均值以及詹氏不等式相关的示例和应用。