On Some Sufficient Conditions for a Function to Be <i>p</i>-Valent Starlike

• Main Authors: Mamoru Nunokawa, Janusz Sokół, Edyta Trybucka
• Format: Article
• Language: English
• Published: MDPI AG 2019-11-01
• Series: Symmetry
• Subjects: univalent functions; starlike; convex; close-to-conve
• Online Access: https://www.mdpi.com/2073-8994/11/11/1417

A function <i>f</i> analytic in a domain <inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mo>&#8712;</mo> <mi mathvariant="double-struck">C</mi> </mrow> </semantics> </math> </inline-formula> is called <i>p</i>-valent in <i>D</i>, if for every complex number <i>w</i>, the equation <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> <mo>=</mo> <mi>w</mi> </mrow> </semantics> </math> </inline-formula> has at most <i>p</i> roots in <i>D</i>, so that there exists a complex number <inline-formula> <math display="inline"> <semantics> <msub> <mi>w</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula> such that the equation <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula> has exactly <i>p</i> roots in <i>D</i>. The aim of this paper is to establish some sufficient conditions for a function analytic in the unit disc <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">D</mi> </semantics> </math> </inline-formula> to be <i>p</i>-valent starlike in <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">D</mi> </semantics> </math> </inline-formula> or to be at most <i>p</i>-valent in <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">D</mi> </semantics> </math> </inline-formula>. Our results are proved mainly by applying Nunokawa&#8217;s lemmas.