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In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D and any integers i,j (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the study of the Bohr phenomenon for a quasisubordination family of functions, and establish the classical Bohr's inequality for the class of quasisubordinate functions. As a consequence, we improve and obtain the exact version of the classical Bohr's inequality for bounded analytic functions and also for K-quasiconformal harmonic mappings by replacing the constant term by the absolute value of the analytic part of the given function. We also obtain the Bohr radius for the subordination family of odd analytic functions. [ABSTRACT FROM AUTHOR]

We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities. For instance, if the matrix \mathbb {A} is partitioned into three blocks A,B,C, then \begin{gather*} |\mathbb {A}|^3 \ge U|A|^3U^* + V|B|^3V^*+ W|C|^3W^*,\\ \sqrt {3} |\mathbb {A}| \ge U|A|U^* + V|B|V^*+ W|C|W^*, \end{gather*} for some isometries U,V,W, and \begin{equation*} \mu _4^2(\mathbb {A}) \le \mu _3^2(A) +\mu _2^2(B) + \mu _1^2(C) \end{equation*} where \mu _j stands for the j-th singular value. Our theorem may be used to extend a result by Bhatia and Kittaneh for the Schatten p-norms and to give a singular value version of Cauchy's Interlacing Theorem. [ABSTRACT FROM AUTHOR]

We consider a Dirichlet problem for the mean curvature operator in the Minkowski spacetime, obtaining a necessary and sufficient condition for the existence of a spacelike solution, with prescribed mean curvature, which is the graph of a function defined on a domain equal to the complement in Rn of the union of a finite number of bounded Lipschitz domains. The mean curvature H = H(x,t) is assumed to have absolute value controlled from above by a locally bounded, Lp-function, p ∈ [1,2n/(n+2)], n ≥ 3. [ABSTRACT FROM AUTHOR]