Related papers: An extension of Nunokawa lemma and its example

Let H[a_0,n] be the class of functions p(z)=a_0+a_nz^n+... which are analytic in the open unit disk U. For p(z)in H[1,2], M. Nunokawa, S. Owa, N. Uyanik and H. Shiraishi (Math. Comput. Modelling. 55 (2012), 1245-1250) have shown some…

Miller and Mocanu proved in their 1997 paper a greatly useful result which is now known as the Open Door Lemma. It provides a sufficient condition for an analytic function on the unit disk to have positive real part. Kuroki and Owa modified…

Let H[a_0,n] be the class of functions f(z)=a_0+a_nz^n+...which are analytic in the open unit disk U}. For f(z) in H[a_0,n], S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65(1978), 289-305) have shown Miller and Mocanu lemma which is…

For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=f"(0)=0, P. T. Mocanu (Mathematica (Cluj), 42(2000)) has considered some sufficient arguments of f'(z)+zf"(z) for |\arg(zf'(z)/f(z))|<\pi\mu/2. The object of the present…

In this paper we give some applications of a lemma of Babalola and Opoola \cite{BO}, which is a classical extension of an earlier one by Lewandowski, Miller and Zlotkiewicz \cite{LMZ}. The applications were given via a new generalization of…

By use of a modified Nunokawa's lemma, we obtain some new conditions for univalence. Also, some sharp inequalities concerning univalent functions are presented.

For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=0, R. Singh and S. Singh (Coll. Math. 47(1982), 309-314) have considered some sufficient problems for f(z) to be univalent in U. The object of the present paper is to…

In the present paper, we obtain a more general conditions for univalence of analytic functions in the open unit disk U. Also, we obtain a refinement to a quasiconformal extension criterion of the main result.

In this note, we consider meromorphic univalent functions $f(z)$ in the unit disc with a simple pole at $z=p\in(0,1)$ which have a $k$-quasiconformal extension to the extended complex plane $\hat{\mathbb C},$ where $0\leq k < 1$. We denote…

There are many results for sufficient conditions of functions f(z) which are analytic in the open unit disc U to be starlike and convex in U. The object of the present paper is to derive some interesting sufficient conditions for f(z) to be…

Let ${\mathcal A}$ be the class of functions analytic in the unit disk ${\mathbb D} := \{ z\in {\mathbb C}:\, |z| < 1 \}$ and normalized such that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we study the class $\mathcal{U}(\lambda)$,…

In this note we consider some generalizations of the Schwarz lemma for harmonic functions on the unit disk, whereby values of such functions and the norms of their differentials at the point $z=0$ are given.

The known Ozaki's condition says that $\mathfrak{Re}\left\{f^{(p)}(z)\right\}>0$ for $|z|<1$ implies that $f(z)=z^p+a_{p+1}z^{p+1}+\cdots$ is at most $p$-valent in $\mathbb D$. In this paper prove an extension of Ozaki's condition. Also, we…

We obtain Schwarz-Pick lemma for $(\alpha, \beta)$-harmonic functions u in the disc, where $\alpha$ and $\beta$ are complex parameters satisfying $\Re \alpha + \Re \beta > -1$. We prove sharp estimate of derivative at the origin for such…

The definition for the $p$-adic Hurwitz-type Euler zeta functions has been given by using the fermionic $p$-adic integral on $\mathbb Z_p$. By computing the values of this kind of $p$-adic zeta function at negative integers, we show that it…

For functions $f(z)=z^p+a_{n+1}z^{p+1}+...$ defined on the open unit disk, the condition $\Re (f'(z)/z^{p-1})>0$ is sufficient for close-to-convexity of $f$. By making use of this result, several sufficient conditions for close-to-convexity…

Let $\mathcal{A}$ denote the class of analytic functions in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ satisfying $f(0)=0$ and $f'(0)=1$. Let $\mathcal{U}$ be the class of functions $f\in\mathcal{A}$ satisfying…

For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=0, P. T. Mocanu (Mathematica (Cluj), 11(34) (1969)) have considered Mocanu functions. The object of the present paper is to discuss some sufficient problems for f(z) to…

In 2002 A.\ Hartmann and X.\ Massaneda obtained necessary and sufficient conditions for interpolation sequences for classes of analytic functions in the unit disc such that $\log M(r,f)=O((1-r)^{-\rho})$, $0<r<1$, $\rho \in (0 , +\infty)$,…

The present essay aims at investigating whether and how far an algebraic analysis of the Zeta Function and of the Riemann Hypothesis can be carried out. Of course the well-established properties of the Zeta Function, explored in depth in…