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Register a new userA new proof of entire function sharing three small functions CM with it $n-$ exact difference
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XiaoHuang Huang
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In this paper, we study the uniqueness of the difference of meromorphic functions. We give a new proof of the following result: Let $f$ be a transcendental meromorphic function of hyper-order less than $1$, let $eta$ be a non-zero complex number, $ngeq1$, an integer, and let $a,b,c$ be three distinct periodic small functions with period $eta$. If $f$ and $Delta_{eta}^{n}f$ share $a,b,c$ CM, then $fequivDelta_{eta}^{n}f$, which using a different method from cite{gkzz}.
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If $f$ is an entire function and $a$ is a complex number, $a$ is said to be an asymptotic value of $f$ if there exists a path $gamma$ from $0$ to infinity such that $f(z) - a$ tends to $0$ as $z$ tends to infinity along $gamma$. The Denjoy--Carleman--Ahlfors Theorem asserts that if $f$ has $n$ distinct asymptotic values, then the rate of growth of $f$ is at least order $n/2$, mean type. A long-standing problem asks whether this conclusion holds for entire functions having $n$ distinct asymptotic (entire) functions, each of growth at most order $1/2$, minimal type. In this paper conditions on the function $f$ and associated asymptotic paths are obtained that are sufficient to guarantee that $f$ satisfies the conclusion of the Denjoy--Carleman--Ahlfors Theorem. In addition, for each positive integer $n$, an example is given of an entire function of order $n$ having $n$ distinct, prescribed asymptotic functions, each of order less than $1/2$.
This paper establishes a version of Nevanlinna theory based on Jackson difference operator $D_{q}f(z)=frac{f(qz)-f(z)}{qz-z}$ for meromorphic functions of zero order in the complex plane $mathbb{C}$. We give the logarithmic difference lemma, the second fundamental theorem, the defect relation, Picard theorem and five-value theorem in sense of Jackson $q$-difference operator. By using this theory, we investigate the growth of entire solutions of linear Jackson $q$-difference equations $D^{k}_{q}f(z)+A(z)f(z)=0$ with meromorphic coefficient $A,$ where $D^k_q$ is Jackson $k$-th order difference operator, and estimate the logarithmic order of some $q$-special functions.
In this paper, we study about existence and non-existence of finite order transcendental entire solutions of the certain non-linear differential-difference equations. We also study about conjectures posed by Rong et al. and Chen et al.
In this paper we shall consider the assymptotic growth of $|P_n(z)|^{1/k_n}$ where $P_n(z)$ is a sequence of entire functions of genus zero. Our results extend a result of J. Muller and A. Yavrian. We shall prove that if the sequence of entire functions has a geometric growth at each point in a set $E$ being non-thin at $infty$ then it has a geometric growth in $CC$ also. Moreover, if $E$ has some more properties, a similar result also holds for a more general kind of growth. Even in the case where $P_n$ are polynomials, our results are new in the sense that it does not require $k_nsucceq deg(P_n)$ as usually required.
In this paper we shall consider the growth at infinity of a sequence $(P_n)$ of entire functions of bounded orders. Our results extend the results in cite{trong-tuyen2} for the growth of entire functions of genus zero. Given a sequence of entire functions of bounded orders $P_n(z)$, we found a nearly optimal condition, given in terms of zeros of $P_n$, for which $(k_n)$ that we have begin{eqnarray*} limsup_{ntoinfty}|P_n(z)|^{1/k_n}leq 1 end{eqnarray*} for all $zin mathbb C$ (see Theorem ref{theo5}). Exploring the growth of a sequence of entire functions of bounded orders lead naturally to an extremal function which is similar to the Siciaks extremal function (See Section 6).
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