We say that an entire function $f\left(z\right)={\sum }_{k=0}{a}_k{z}^{n_k}(0=n_0\<n_1\<n_2\<...)$ has Fejér gaps if ${\sum }_{k=1}^{\infty }1/n_k\<\infty .$ The main result of this paper is as follows: An entire function with Fejér gaps has no finite deficient value.

Se estudia el término de error en el segundo teorema fundamental para las aplicaciones holomorfas F : Y → P1, donde en Y hay definida una aplicación recubridora holomorfa p : Y → C, aplicaciones consideradas por S. Lang and W. Cherry. También se obtiene el lema de la derivada logarítmica para esta clase de funciones que, en particular, contiene la clase de las funciones algebroides. Se demuestra que la estimación obtenida mejora un resultado clásico de Henrik Selberg sobre la derivada logarítmica...

This paper is devoted to considering the iterated order and the fixed points of some differential polynomials generated by solutions of the differential equation $$f^{{}^{\text{'}\text{'}}}+A_1\left(z\right)f^{{}^{\text{'}}}+A_0\left(z\right)f=F,$$ where $A_1\left(z\right)$, $A_0\left(z\right)$$(\neg \equiv 0)...$

The principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep...

Suppose that $f\left(z\right)$ is a meromorphic or entire function satisfying $P\left(z,f,f^{\prime },\mathrm{\dots },f^{\left(n\right)}\right)=0$ where $P$ is a polynomial in all its arguments. Is there a limitation on the growth of $f$, as measured by its characteristic $T\left(r,f\right)$? In general the answer to this question is not known. Theorems of Gol'dberg, Steinmetz and the author give a positive answer in certain cases. Some illustrative examples are also given.

We prove some theorems on the hyper-order of solutions of the equation $f^{\left(k\right)}-e^{Q}f=a(1-e^Q)$, where Q is an entire function, which is a polynomial or not, and a is an entire function whose order can be larger than 1. We improve some results by J. Wang and X. M. Li.

The purpose of this paper is to establish a theorem which answers a conjecture of Paley on the distribution of values of Hadamard lacunary series and which is useful to study the Peano curve property of such series.

We consider the zero distribution of difference-differential polynomials of meromorphic functions and present some results which can be seen as the discrete analogues of the Hayman conjecture. In addition, we also investigate the uniqueness of difference-differential polynomials of entire functions sharing one common value. Our theorems improve some results of Luo and Lin [J. Math. Anal. Appl. 377 (2011), 441-449] and Liu, Liu and Cao [Appl. Math. J. Chinese Univ. 27 (2012), 94-104].