On an extension of Nunokawa’s lemma

Jack’s Lemma says that if f(z) is regular in the disc $$|z|\le r$$ | z | ≤ r , $$f(0)=0$$ f ( 0 ) = 0 , and |f(z)| assumes its maximum at $$z_0$$ z 0 on the circle $$|z|=r$$ | z | = r , then $$z_0f'(z)_0/f(z_0)\ge 1$$ z 0 f ′ ( z ) 0 / f ( z 0 ) ≥ 1 . This Lemma was generalized in several directions. In this paper we consider an improvement of some first author’s results of this type.

Certain sufficient conditions for close-to-convexity and starlikeness of multivalent functions

By using Jack's lemma, we derive simple sufficient conditions for analytic functions to be multivalent close-to-convex and multivalent starlike.

Applications of the Jack's lemma for the meromorphic functions at the boundary

In this paper, a boundary version of the Schwarz lemma for the class $\mathcal{% N(\alpha )}$ is investigated. For the function $f(z)=\frac{1}{z}% +a_{0}+a_{1}z+a_{2}z^{2}+...$ defined in the punctured disc $E$ such that $% f(z)\in \mathcal{N(\alpha )}$, we estimate a modulus of the angular derivative of the function $\frac{zf^{\prime }(z)}{f(z)}$ at the boundary point $c$ with $\frac{cf^{\prime }(c)}{f(c)}=\frac{1-2\beta }{\beta }$. Moreover, Schwarz lemma for class $\mathcal{N(\alpha )}$ is given.

On a local version of Jack’s lemma

The purpose of this paper is to provide a result which concerns with the boundary behavior of analytic functions. It may be a local version of the well known Jack’s lemma when we change the function normalization at the origin.

On a new proof and an extension of Jack’s lemma

We give a new proof and discuss an extension of Jack’s lemma for polynomials.