On the Strong Starlikeness of the Bernardi Transform

oleh: Zahra Orouji, Ali Ebadian, Nak Eun Cho

Format: Article
Diterbitkan: MDPI AG 2023-01-01

Deskripsi

Many papers concern both the starlikeness and the convexity of Bernardi integral operator. Using the Nunokawa’s Lemma, we want to determine conditions for the strong starlikeness of the Bernardi transform of normalized analytic functions <i>g</i>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>|</mo><mo form="prefix">arg</mo></mrow><mrow><mo>{</mo><msup><mi>g</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>}</mo></mrow><mrow><mo>|</mo><mo> </mo><mo><</mo></mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>α</mi><mi>π</mi></mrow><mn>2</mn></mfrac></mstyle></mrow></semantics></math></inline-formula> in the open unit disk <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>. Our results include the results of Mocanu, Nunokawa and others on the Libera transform.