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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 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]

Let K/Q be an abelian extension and let D be the absolute value of the discriminant of K. We show that for each e > 0, the smallest rational prime that splits completely in K is O(D1/4+e). Here the implied constant depends only on e and the degree of K. This generalizes a theorem of Elliott, who treated the case when K/Q has prime conductor. [ABSTRACT FROM AUTHOR]