Entire Function Of Genus Research Articles

Extension of Inverses of Entire Functions of Genus 1 and 2 to the Upper Half Plane

Any entire function of genus 1 which is positive on the positive real axis and which has only negative zeros decreases on some unbounded interval of the positive axis. The inverse of its reciprocal is shown to have an extension from that interval to a Pick-function in the upper half plane. A similar result holds for a class of entire functions of genus 2 and in particular the inverse function of Barnes’ double gamma function on a certain interval of the positive axis can be extended to a Pick-function. These results are proved using positive and negative definite kernels.

Sums of exponential functions having only real zeros

Let Hn(z) be the function of a complex variable z defined by where the summation is over all 2n possible plus and minus sign combinations, the same sign combination being used in both the argument of G and in the exponent. The numbers \({{a_1,a_2,a_3,\ldots}}\) and \({{b_1,b_2,b_3,\ldots}}\) are assumed to be positive, and G is an entire function of genus 0 or 1 that is real on the real axis and has only real zeros. Then the function Hn(z) has only real zeros.

URS' For Weierstrass Products without Exponential Factors

Let $ \\cal W $ be the set of entire functions equal to a Weierstrass product of the form $ {f(x)= Ax^q\\lim_{r \ o \\infty} \\prod_{|a_j|\\leq r}{(1- \\fraca {x} {a_j})}} $ where the convergence is uniform in all bounded subsets of $ {\\shadC} $ , let $ \\cal V $ be the set of $ f\\in {\\cal W} $ such that $ {\\shadC} [\\,f]\\subset {\\cal W} $ , and let $ {\\cal H} $ be the $ {\\shadC} $ -algebra of entire functions satisfying $ { {\\lim_{r\ o \\infty } } ({\\ln M(r,f) / r})=0} $ . Then $ \\cal H $ is included in $ {\\cal V} $ and strictly contains the set of entire functions of genus zero, (which, itself, strictly contains the $ {\\shadC} $ -algebra of entire functions of order 𝜌 < 1). Let $ n, m\\in {\\shadN} ^* $ satisfy n > m S 3. Let $ a\\in {\\shadC}^* $ satisfies $ {a^n\ ot = \\fraca{n^n}{(m^m(n-m)^{n-m}})} $ and assume that for every ( n m m )-th root ξ of 1 different from m 1, a satisfies further $ {a^{n}\ eq (1+\\xi )^{n-m} (\\fraca{n^n}{((n-m)^{n-m}m^m}))} $ . Let P ( X ) = X n m aX m + 1 and let T n,m ( a ) be the set of its zeros. Then T n,m ( a ) has n distinct points and is a urs for $ {\\cal V} $ . In particular this applies to functions such as sin x and cos x .

Non-real Zeros of Derivatives of Real Entire Functions and the Pólya-Wiman Conjectures

A function f is in the class $ V_2p $ iff $ f(z) = e^{-az^{2p+2}}g(z) $ where a S 0 and g is a constant multiple of a real entire function of genus h 2 p + 1 with only real zeros. The class $ U_2p $ is defined as follows: $ U_0 = V_0 $ , $ U_{2p} = V_{2p}-V_{2p-2} $ . Functions in the class $ U_{2p}^{*} $ are represented as $ g(z) = c(z)f(z) $ where $ f\\in U_{2p} $ and c is a real polynomial with no real zeros. Every real entire function g , of finite order with at most finitely many non-real zeros satisfies $ g\\in U_{2p}^{*} $ for a unique p . We show the exact number of non-real zeros of f" , for $ f\\in U_{2p} $ , in terms of the number of non-real zeros of f' and a geometrical condition on the components of Im Q ( z ) > 0, where $ \\displaystyle Q(z) = z-({f(z)}/{f'(z)}) $ . Further, for a subclass of $ f\\in U_{2p} $ , we show necessary and sufficient conditions for f" to have exactly 2 p non-real zeros. For a subclass of $ U_{2p}^{*} $ we show that if f' has only real zeros, then f" has exactly 2 p non-real zeros. For $ f\\in U_{2p}^{*} $ we show that 2 p is a lower bound for the number of non-real zeros of $ f^{(k)} $ for k S 2.

On a theorem of Craven, Csordas and Smith

We get a simple and direct proof of the following known result. Let f(z)-eαz2 g(z) where α← 0 and g is a real entire function of genus at most one. If f has only a finite number of nonreal zeros, then its derivatives, from a certain one onward, have only real zeros.

Minimum of the modulus of an entire function on the sequence of Polya peaks

Let f be an entire function of genus zero, let (rk) be a sequence of Polya peaks for N(r, f) of order λ<1. Then there exists a sequence rk′~rk such that If for (rk) one takes a sequence of Polya peaks for ln M(r, f) or for T(r, f), 1/2<λ<1, then the result ceases to be true.

On a theorem of S. Bernstein

S. Bernstein (see(l), pp. 198–204) has proved that ifis an entire function of genus zero, then the seriesis convergent. By considering the most unfavourable case, when all the (βn) are positive real numbers, and writingwhich is convergent by hypothesis, he proved that.