In this note, we consider a holomorphic mapping f from the unit disk C in C to p-ball B^p = {z∈C^n;i=1∑n|zi|p〈1,1〈p〈+∞. It is proved that for such f,| | |f||(z)|≤1-||f(z)||^2/1-|z|^2,z∈D. Th...In this note, we consider a holomorphic mapping f from the unit disk C in C to p-ball B^p = {z∈C^n;i=1∑n|zi|p〈1,1〈p〈+∞. It is proved that for such f,| | |f||(z)|≤1-||f(z)||^2/1-|z|^2,z∈D. The extremal problem is also discussed when p is an even number. This result extends some related results on Schwarz lemma.展开更多
Let|Bnp|,1<p<∞ , be the volume of the unit p-ball in Rn and q the H?lder conjugate exponent of p. We represent the volume product |Bnp| |Bna| as a function free of its gamma symbolism. This representation will ...Let|Bnp|,1<p<∞ , be the volume of the unit p-ball in Rn and q the H?lder conjugate exponent of p. We represent the volume product |Bnp| |Bna| as a function free of its gamma symbolism. This representation will allows us in this particular case to confirm, using basic classical analysis tools, two conjectured and partially proved lower and upper bounds for the volume product of centrally symmetric convex bodies of the Euclidean Rn . These bounds in the general case play a central role in convex geometric analysis.展开更多
基金supported by National Natural Science Foundations of China(11011373,11201199,11271333)Zhejiang Provincial Natural Science Foundation of China(LY14A010008)
文摘In this note, we consider a holomorphic mapping f from the unit disk C in C to p-ball B^p = {z∈C^n;i=1∑n|zi|p〈1,1〈p〈+∞. It is proved that for such f,| | |f||(z)|≤1-||f(z)||^2/1-|z|^2,z∈D. The extremal problem is also discussed when p is an even number. This result extends some related results on Schwarz lemma.
文摘Let|Bnp|,1<p<∞ , be the volume of the unit p-ball in Rn and q the H?lder conjugate exponent of p. We represent the volume product |Bnp| |Bna| as a function free of its gamma symbolism. This representation will allows us in this particular case to confirm, using basic classical analysis tools, two conjectured and partially proved lower and upper bounds for the volume product of centrally symmetric convex bodies of the Euclidean Rn . These bounds in the general case play a central role in convex geometric analysis.