Fourth Hankel determinant and logarithmic coefficients for starlike functions associated with the cosine function
UDC 517.53, 517.54 Let $\mathcal{\mathcal{S}}_{\cos}^{\ast}$ be a class of normalized analytic functions $f$ in an open unit disk $\mathbb{D}$ satisfying the subordination $\dfrac{zf^{\prime}(z)}{f(z)}\prec\cos z.$ The aim of the present article is to find upper bounds for the module of some coeffic...
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| Дата: | 2026 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2026
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/9403 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1866663722546626560 |
|---|---|
| author | Ali, Rashid Raza, Mohsan Bulboacă, Teodor Ali, Rashid Raza, Mohsan Bulboacă, Teodor |
| author_facet | Ali, Rashid Raza, Mohsan Bulboacă, Teodor Ali, Rashid Raza, Mohsan Bulboacă, Teodor |
| author_institution_txt_mv | [
{
"author": "Rashid Ali",
"institution": "Department of Mathematics, Government College University Faisalabad, Pakistan"
},
{
"author": "Mohsan Raza",
"institution": "Department of Mathematics, Government College University Faisalabad, Pakistan"
},
{
"author": "Teodor Bulboacă",
"institution": "Faculty of Mathematics and Computer Science, Research Center of Applied Analysis, Babeş-Bolyai University, Cluj-Napoca, Romania"
}
] |
| author_sort | Ali, Rashid |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-05-30T12:43:39Z |
| description | UDC 517.53, 517.54
Let $\mathcal{\mathcal{S}}_{\cos}^{\ast}$ be a class of normalized analytic functions $f$ in an open unit disk $\mathbb{D}$ satisfying the subordination $\dfrac{zf^{\prime}(z)}{f(z)}\prec\cos z.$ The aim of the present article is to find upper bounds for the module of some coefficients, of the fourth Hankel determinant $H_{4,1}(f)$ for the function class $\mathcal{\mathcal{S}}_{\cos}^{\ast},$ and of some functionals defined by using the expansion coefficients in Taylor series. Moreover, we also find the upper bounds for the fifth and sixth logarithmic coefficients obtained for the functions from the same class. Note that some of our results are the best possible. The results of our paper improve numerous versions of the results recently presented in [K. Marimuthu, J. Uma, and T. Bulboacă, Hacet. J. Math. Stat., 52, No. 3, 596 (2023)]. The tools used in the proofs have been recently obtained in [N. E. Cho, B. Kowalczyk, A. Lecko, and B. ´Śmiarowska, Filomat, 34, No. 6, 2061 (2020)]. They are combined with the results from [F. Carlson, Ark. Mat. Astr. Fys., 27A, No. 1, 8 (1939)], and with the method aimed at finding the extrema of real functions of many variables. |
| doi_str_mv | 10.3842/umzh.v78i5-6.9403 |
| first_indexed | 2026-05-30T01:00:56Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-9403 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-05-31T01:00:49Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-94032026-05-30T12:43:39Z Fourth Hankel determinant and logarithmic coefficients for starlike functions associated with the cosine function Fourth Hankel determinant and logarithmic coefficients for starlike functions associated with the cosine function Ali, Rashid Raza, Mohsan Bulboacă, Teodor Ali, Rashid Raza, Mohsan Bulboacă, Teodor Hankel determinants, Fekete-Szeg\H{o} functional logarithmic coefficients Starlike functions 30C45 30C50 30C80 UDC 517.53, 517.54 Let $\mathcal{\mathcal{S}}_{\cos}^{\ast}$ be a class of normalized analytic functions $f$ in an open unit disk $\mathbb{D}$ satisfying the subordination $\dfrac{zf^{\prime}(z)}{f(z)}\prec\cos z.$ The aim of the present article is to find upper bounds for the module of some coefficients, of the fourth Hankel determinant $H_{4,1}(f)$ for the function class $\mathcal{\mathcal{S}}_{\cos}^{\ast},$ and of some functionals defined by using the expansion coefficients in Taylor series. Moreover, we also find the upper bounds for the fifth and sixth logarithmic coefficients obtained for the functions from the same class. Note that some of our results are the best possible. The results of our paper improve numerous versions of the results recently presented in [K. Marimuthu, J. Uma, and T. Bulboacă, Hacet. J. Math. Stat., 52, No. 3, 596 (2023)]. The tools used in the proofs have been recently obtained in [N. E. Cho, B. Kowalczyk, A. Lecko, and B. ´Śmiarowska, Filomat, 34, No. 6, 2061 (2020)]. They are combined with the results from [F. Carlson, Ark. Mat. Astr. Fys., 27A, No. 1, 8 (1939)], and with the method aimed at finding the extrema of real functions of many variables. УДК 517.53, 517.54 Четвертий визначник Ганкеля та логарифмічні коефіцієнти для зіркоподібних функцій, що пов'язані з косинусом Позначимо через $\mathcal{\mathcal{S}}_{\cos}^{\ast}$ клас нормалізованих аналітичних функцій $f$ у відкритому одиничному диску $\mathbb{D},$ які задовольняють підпорядкування $\dfrac{zf^{\prime}(z)}{f(z)}\prec\cos z.$ Метою цієї статті є знаходження верхньої межі для модуля деяких коефіцієнтів і четвертого визначника Ганкеля $H_{4,1}(f)$ для класу функцій $\mathcal{\mathcal{S}}_{\cos}^{\ast},$ а також для деяких функціоналів, визначених через коефіцієнти розкладу в ряд Тейлора. Крім того, також одержано верхні межі для п'ятого та шостого логарифмічних коефіцієнтів для функцій з цього класу. Більш того, деякі з отриманих результатів є найкращими з можливих. Результати статті вдосконалюють низку результатів, представлених нещодавно у [K. Marimuthu, J. Uma, T. Bulboacă, Hacet. J. Math. Stat., 52, No. 3, 596 (2023)]. Інструменти, що були використані в доведеннях, були отримані нещодавно у [N. E. Cho, B. Kowalczyk, A. Lecko, B. ´Śmiarowska, Filomat, 34, No. 6, 2061 (2020)] у поєднанні з результатами з [F. Carlson, Ark. Mat. Astr. Fys., 27A, No. 1, 8 (1939)] і методом знаходження екстремуму дійсних функцій багатьох змінних. Institute of Mathematics, NAS of Ukraine 2026-05-29 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/9403 10.3842/umzh.v78i5-6.9403 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 5-6 (2026); 359–360 Український математичний журнал; Том 78 № 5-6 (2026); 359–360 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/9403/10652 Copyright (c) 2026 Rashid Ali, Mohsan Raza, Teodor Bulboacă |
| spellingShingle | Ali, Rashid Raza, Mohsan Bulboacă, Teodor Ali, Rashid Raza, Mohsan Bulboacă, Teodor Fourth Hankel determinant and logarithmic coefficients for starlike functions associated with the cosine function |
| title | Fourth Hankel determinant and logarithmic coefficients for starlike functions associated with the cosine function |
| title_alt | Fourth Hankel determinant and logarithmic coefficients for starlike functions associated with the cosine function |
| title_full | Fourth Hankel determinant and logarithmic coefficients for starlike functions associated with the cosine function |
| title_fullStr | Fourth Hankel determinant and logarithmic coefficients for starlike functions associated with the cosine function |
| title_full_unstemmed | Fourth Hankel determinant and logarithmic coefficients for starlike functions associated with the cosine function |
| title_short | Fourth Hankel determinant and logarithmic coefficients for starlike functions associated with the cosine function |
| title_sort | fourth hankel determinant and logarithmic coefficients for starlike functions associated with the cosine function |
| topic_facet | Hankel determinants Fekete-Szeg\H{o} functional logarithmic coefficients Starlike functions 30C45 30C50 30C80 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9403 |
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