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Register a new userExtreme values of the derivative of Blaschke products and hypergeometric polynomials
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A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke products: those that maximize the difference between the maximum and minimum of the derivative, and those that minimize it. Both classes turn out to have the same algebraic structure, being the quotient of two hypergeometric polynomials.
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The determination of a finite Blaschke product from its critical points is a well-known problem with interrelations to other topics. Though existence and uniqueness of solutions are established for long, we present several new aspects which have not yet been explored to their full extent. In particular, we show that the following three problems are equivalent: (i) determining a finite Blaschke product from its critical points, (ii) finding the equilibrium position of moveable point charges interacting with a special configuration of fixed charges, (iii) solving a moment problem for the canonical representation of power moments on the real axis. These equivalences are not only of theoretical interest, but also open up new perspectives for the design of algorithms. For instance, the second problem is closely linked to the determination of certain Stieltjes and Van Vleck polynomials for a second order ODE and allows the description of solutions as global minimizers of an energy functional.
In this paper we shall use the boundary Schwarz lemma of Osserman to obtain some generalizations and refinements of some well known results concerning the maximum modulus of the polynomials with restricted zeros due to Turan, Dubinin and others.
We will provide sufficient conditions for the shifted hypergeometric function $z_2F_1(a,b;c;z)$ to be a member of a specific subclass of starlike functions in terms of the complex parameters $a,b$ and $c.$ For example, we study starlikeness of order $alpha,$ $lambda$-spirallikeness of order $alpha$ and strong starlikeness of order $alpha.$ In particular, those properties lead to univalence of the shifted hypergeometric functions on the unit disk.
This note investigates the prime values of the polynomial $f(t)=qt^2+a$ for any fixed pair of relatively prime integers $ ageq 1$ and $ qgeq 1$ of opposite parity. For a large number $xgeq1$, an asymptotic result of the form $sum_{nleq x^{1/2},, n text{ odd}}Lambda(qn^2+a)gg qx^{1/2}/2varphi(q)$ is achieved for $qll (log x)^b$, where $ bgeq 0 $ is a constant.
A recent construction by Amarra, Devillers and Praeger of block designs with specific parameters depends on certain quadratic polynomials, with integer coefficients, taking prime power values. The Bunyakovsky Conjecture, if true, would imply that each of them takes infinitely many prime values, giving an infinite family of block designs with the required parameters. We have found large numbers of prime values of these polynomials, and the numbers found agree very closely with the estimates for them provided by Lis recent modification of the Bateman-Horn Conjecture. While this does not prove that these polynomials take infinitely many prime values, it provides strong evidence for this, and it also adds extra support for the validity of the Bunyakovsky and Bateman-Horn Conjectures.
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