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Register a new userGeneral concavity of minimal $L^{2}$ integrals related to multiplier ideal sheaves
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Qian Guan
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In this note, we present a general version of the concavity of the minimal $L^{2}$ integrals related to multiplier ideal sheaves.
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In this note, we present the concavity of the minimal $L^2$ integrals related to multiplier ideals sheaves on Stein manifolds. As applications, we obtain a necessary condition for the concavity degenerating to linearity, a characterization for 1-dimensional case, and a characterization for the equality in 1-dimensional optimal $L^{2}$ extension problem to hold.
In this note, we reveal that our solution of Demaillys strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Koll{a}r and Jonsson-Mustatu{a} implies the truth of twist
Chen proposed a conjecture on the log-concavity of the generating function for the symmetric group with respect to the length of longest increasing subsequences of permutations. Motivated by Chens log-concavity conjecture, B{o}na, Lackner and Sagan further studied similar problems by restricting the whole symmetric group to certain of its subsets. They obtained the log-concavity of the corresponding generating functions for these subsets by using the hook-length formula. In this paper, we generalize and prove their results by establishing the Schur positivity of certain symmetric functions. This also enables us to propose a new approach to Chens original conjecture.
In this note, we answer a question on the extension of $L^{2}$ holomorphic functions posed by Ohsawa.
In the present paper the new multiplier transformations $mathrm{{mathcal{J}% }}_{p}^{delta }(lambda ,mu ,l)$ $(delta ,lgeq 0,;lambda geq mu geq 0;;pin mathrm{% }%mathbb{N} )}$ of multivalent functions is defined. Making use of the operator $mathrm{% {mathcal{J}}}_{p}^{delta }(lambda ,mu ,l),$ two new subclasses $mathcal{% P}_{lambda ,mu ,l}^{delta }(A,B;sigma ,p)$ and $widetilde{mathcal{P}}% _{lambda ,mu ,l}^{delta }(A,B;sigma ,p)$textbf{ }of multivalent analytic functions are introduced and investigated in the open unit disk. Some interesting relations and characteristics such as inclusion relationships, neighborhoods, partial sums, some applications of fractional calculus and quasi-convolution properties of functions belonging to each of these subclasses $mathcal{P}_{lambda ,mu ,l}^{delta }(A,B;sigma ,p)$ and $widetilde{mathcal{P}}_{lambda ,mu ,l}^{delta }(A,B;sigma ,p)$ are investigated. Relevant connections of the definitions and results presented in this paper with those obtained in several earlier works on the subject are also pointed out.
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