scholarly journals On an extension of Nunokawa’s lemma

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Mamoru Nunokawa ◽  
Janusz Sokół

AbstractJack’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.

2019 ◽  
Vol 38 (7) ◽  
pp. 219-226
Author(s):  
Tugba Akyel ◽  
Bulent Nafi Ornek

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.


2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Saibah Siregar ◽  
Maslina Darus

For , , we consider the of normalized analytic convex functions defined in the open unit disc . In this paper, we investigate the class , that is, , with is Koebe type, that is, . The subordination result for the aforementioned class will be given. Further, by making use of Jack's Lemma as well as several differential and other inequalities, the authors derived sufficient conditions for starlikeness of the class of -fold symmetric analytic functions of Koebe type. Relevant connections of the results presented here with those given in the earlier works are also indicated.


2020 ◽  
Vol 23 (2) ◽  
pp. 201-210
Author(s):  
Mohamed K. Aouf ◽  
Teodor Bulboacă ◽  
Adela O. Mostafa

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


2018 ◽  
Vol 48 (2) ◽  
pp. 125-139
Author(s):  
Bülent Nafi Örnek ◽  
Selin Aydınoğlu

2019 ◽  
Vol 49 (6) ◽  
pp. 1869-1875
Author(s):  
Richard Fournier
Keyword(s):  

2017 ◽  
Vol 23 (1) ◽  
Author(s):  
Richard Fournier
Keyword(s):  

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


2007 ◽  
Vol 7 (2) ◽  
pp. 371-378 ◽  
Author(s):  
Richard Fournier ◽  
Marius Serban

2017 ◽  
Author(s):  
Zainab E. Abdulnaby ◽  
Rabha W. Ibrahim ◽  
Adem Kılıçman

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