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Register a new userOn certain type of difference polynomials of meromorphic functions
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In this paper, we investigate zeros of difference polynomials of the form $f(z)^nH(z, f)-s(z)$, where $f(z)$ is a meromorphic function, $H(z, f)$ is a difference polynomial of $f(z)$ and $s(z)$ is a small function. We first obtain some inequalities for the relationship of the zero counting function of $f(z)^nH(z, f)-s(z)$ and the characteristic function and pole counting function of $f(z)$. Based on these inequalities, we establish some difference analogues of a classical result of Hayman for meromorphic functions. Some special cases are also investigated. These results improve previous findings.
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In this paper we extend the concept of bi-univalent to the class of meromorphic functions. We propose to investigate the coefficient estimates for two classes of meromorphic bi-univalent functions. Also, we find estimates on the coefficients |b0| and |b1| for functions in these new classes. Some interesting remarks and applications of the results presented here are also discussed.
In this paper, we study the uniqueness of meromporphic functions and their difference operators. In particular, We have proved: Let $f$ be a nonconstant entire function on $mathbb{C}^{n}$, let $etain mathbb{C}^{n}$ be a nonzero complex number, and let $a$ and $b$ be two distinct complex numbers in $mathbb{C}^{n}$. If $$varlimsup_{rrightarrowinfty}frac{logT(r,f)}{r}=0,$$ and if $f(z)$ and $(Delta_{eta}^{n}f(z))^{(k)}$ share $a$ CM and share $b$ IM, then $f(z)equiv(Delta_{eta}^{n}f(z))^{(k)}$.
We introduce the class of analytic functions $$mathcal{F}(psi):= left{fin mathcal{A}: left(frac{zf(z)}{f(z)}-1right) prec psi(z),; psi(0)=0 right},$$ where $psi$ is univalent and establish the growth theorem with some geometric conditions on $psi$ and obtain the Koebe domain with some related sharp inequalities. Note that functions in this class may not be univalent. As an application, we obtain the growth theorem for the complete range of $alpha$ and $beta$ for the functions in the classes $mathcal{BS}(alpha):= {fin mathcal{A} : ({zf(z)}/{f(z)})-1 prec {z}/{(1-alpha z^2)},; alphain [0,1) }$ and $mathcal{S}_{cs}(beta):= {fin mathcal{A} : ({zf(z)}/{f(z)})-1 prec {z}/({(1-z)(1+beta z)}),; betain [0,1) }$, respectively which improves the earlier known bounds. The sharp Bohr-radii for the classes $S(mathcal{BS}(alpha))$ and $mathcal{BS}(alpha)$ are also obtained. A few examples as well as certain newly defined classes on the basis of geometry are also discussed.
In [Israel J. Math, 2014], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions $mathcal F$ in a domain $Dsubset mathbb C,$ and for a positive constant $epsilon$, if for each $fin mathcal F$ there exist meromorphic functions $a_f,b_f,c_f$ such that $f$ omits $a_f,b_f,c_f$ in $D$ and $$min{rho(a_f(z),b_f(z)), rho(b_f(z),c_f(z)), rho(c_f(z),a_f(z))}geq epsilon,$$ for all $zin D$, then $mathcal F$ is normal in $D$. Here, $rho$ is the spherical metric in $widehat{mathbb C}$. In this paper, we establish the high-dimension
We give conditions characterizing holomorphic and meromorphic functions in the unit disk of the complex plane in terms of certain weak forms of the maximum principle. Our work is directly inspired by recent results of John Wermer, and by the theory of the projective hull of a compact subset of complex projective space developed by Reese Harvey and Blaine Lawson.
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