On the Strong Starlikeness of the Bernardi Transform
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...
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MDPI AG
2023-01-01
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| Online Access: | https://www.mdpi.com/2075-1680/12/1/91 |
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| author | Zahra Orouji Ali Ebadian Nak Eun Cho |
| author_facet | Zahra Orouji Ali Ebadian Nak Eun Cho |
| author_sort | Zahra Orouji |
| collection | DOAJ |
| description | 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. |
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| last_indexed | 2024-03-09T13:35:55Z |
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| spelling | doaj.art-46147917fa644c76b4813cef0c4a92102023-11-30T21:12:12ZengMDPI AGAxioms2075-16802023-01-011219110.3390/axioms12010091On the Strong Starlikeness of the Bernardi TransformZahra Orouji0Ali Ebadian1Nak Eun Cho2Department of Mathematics, Faculty of Science, Urmia University, Urmia 5756151818, IranDepartment of Mathematics, Faculty of Science, Urmia University, Urmia 5756151818, IranDepartment of Applied Mathematics, College of Natural Sciences, Pukyong National University, Busan 608-737, Republic of KoreaMany 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.https://www.mdpi.com/2075-1680/12/1/91Nunokawa’s lemmaLibera operatorBernardi operatorstrongly starlike functions |
| spellingShingle | Zahra Orouji Ali Ebadian Nak Eun Cho On the Strong Starlikeness of the Bernardi Transform Axioms Nunokawa’s lemma Libera operator Bernardi operator strongly starlike functions |
| title | On the Strong Starlikeness of the Bernardi Transform |
| title_full | On the Strong Starlikeness of the Bernardi Transform |
| title_fullStr | On the Strong Starlikeness of the Bernardi Transform |
| title_full_unstemmed | On the Strong Starlikeness of the Bernardi Transform |
| title_short | On the Strong Starlikeness of the Bernardi Transform |
| title_sort | on the strong starlikeness of the bernardi transform |
| topic | Nunokawa’s lemma Libera operator Bernardi operator strongly starlike functions |
| url | https://www.mdpi.com/2075-1680/12/1/91 |
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