Öz
In this paper, a novel subclass, denoted as $\mathcal{PH}(q, \alpha)$, is unveiled within the domain of harmonic functions in the open unit disk $\mathbb{E}$. This subclass, comprised of functions $\mathfrak{f}=\mathfrak{u}+\overline{\mathfrak{v}}\in \mathcal{SH}^{0}$, is characterized by a specific inequality involving the $q$-derivative operator. Through meticulous analysis, it is demonstrated that functions belonging to $\mathcal{PH}(q, \alpha)$ exhibit remarkable close-to-convexity properties. Furthermore, diverse results such as distortion theorem, coefficient bounds, and a sufficient coefficient condition are yielded by the exploration. Additionally, the closure properties of $\mathcal{PH}(q, \alpha)$ under convolution operations and convex combination are elucidated, underscoring its structural coherence and relevance in the broader context of harmonic mappings.
Anahtar Kelimeler
q-Derivative harmonic functions q-close-to-convex functions coefficient bounds distortion.
Kaynakça
- Ahuja, O.P., C¸ etinkaya, A., Use of Qquantum calculus approach in mathematical sciences and its role in geometric function theory, AIP Conference Proceedings, 2095(2019), 020001-14.
- Ahuja, O.P., C¸ etinkaya, A., Connecting quantum calculus and harmonic starlike functions, Filomat, 34(5)(2020), 1431–1441.
- Ahuja, O.P., C¸ etinkaya, A., Polatoglu, Y., Harmonic univalent convex functions using a quantum calculus approach, Acta Universitatis Apulensis, 58(2019), 67–81.
- Çakmak, S., Yaşar, E., Yalçın, S., New subclass of the class of close-to-convex harmonic mappings defined by a third-order differential inequality, Hacettepe Journal of Mathematics and Statistics, 51(1)(2022), 172–186.
- Çakmak, S., Regarding a novel subclass of harmonic multivalent functions defined by higher-order differential inequality, Iranian Journal of Science, 48(2024), 1541–1550.
- Cakmak, S., Yasar, E., Yalc¸ın Tokgöz, S., Some basic properties of a subclass of close-to-convex harmonic mappings, Turkic World Mathematical Society (TWMS) Journal of Pure and Applied Mathematics, 15(2)(2024), 163–173.
- Clunie, J., Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I., 9(1984), 3–25.
- Feje´r, L., Über die Positivita¨t von Summen, die nach trigonometrischen oder Legendreschen Funktionen fortschreiten, Acta Litt. Ac Sei. Szeged, (1925), 75–86.
- Goodman, A.W., Univalent Functions, vol. I, Mariner Publishing Co., Inc., Tampa, FL, 1983.
- Jackson, F.H., On q-functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46(2)(1909), 253–281.
- Owa, S., Nunokawa, M., Saitoh, H., Srivastava, H.M., Close-to-convexity, starlikeness, and convexity of certain analytic functions, Appl. Math. Lett., 15(1)(2002), 63–69.
- Ponnusamy, S., Yamamoto, H., Yanagihara, H., Variability regions for certain families of harmonic univalent mappings, Complex Var. Elliptic Equ., 58(1)(2013), 23–34.
- Li, L., Ponnusamy, S., Disk of convexity of sections of univalent harmonic functions, J. Math. Anal. Appl., 408(2013), 589–596.
- Li, L., Ponnusamy, S., Injectivity of sections of univalent harmonic mappings, Nonlinear Anal., 89(2013), 276–283.
- Salagean, G.S., Subclass of univalent functions, in Complex Analysis-Fifth Romanian Finish Seminar, (1983), 362–372.
- Singh, R., Singh, S., Convolution properties of a class of starlike functions, Proc. Amer. Math. Soc., 106(1989), 145–152.
- Yalçın, S., Bayram, H., On harmonic univalent functions involving q-poisson distribution series, MJPS, 8(2)(2021).
Öz
Kaynakça
- Ahuja, O.P., C¸ etinkaya, A., Use of Qquantum calculus approach in mathematical sciences and its role in geometric function theory, AIP Conference Proceedings, 2095(2019), 020001-14.
- Ahuja, O.P., C¸ etinkaya, A., Connecting quantum calculus and harmonic starlike functions, Filomat, 34(5)(2020), 1431–1441.
- Ahuja, O.P., C¸ etinkaya, A., Polatoglu, Y., Harmonic univalent convex functions using a quantum calculus approach, Acta Universitatis Apulensis, 58(2019), 67–81.
- Çakmak, S., Yaşar, E., Yalçın, S., New subclass of the class of close-to-convex harmonic mappings defined by a third-order differential inequality, Hacettepe Journal of Mathematics and Statistics, 51(1)(2022), 172–186.
- Çakmak, S., Regarding a novel subclass of harmonic multivalent functions defined by higher-order differential inequality, Iranian Journal of Science, 48(2024), 1541–1550.
- Cakmak, S., Yasar, E., Yalc¸ın Tokgöz, S., Some basic properties of a subclass of close-to-convex harmonic mappings, Turkic World Mathematical Society (TWMS) Journal of Pure and Applied Mathematics, 15(2)(2024), 163–173.
- Clunie, J., Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I., 9(1984), 3–25.
- Feje´r, L., Über die Positivita¨t von Summen, die nach trigonometrischen oder Legendreschen Funktionen fortschreiten, Acta Litt. Ac Sei. Szeged, (1925), 75–86.
- Goodman, A.W., Univalent Functions, vol. I, Mariner Publishing Co., Inc., Tampa, FL, 1983.
- Jackson, F.H., On q-functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46(2)(1909), 253–281.
- Owa, S., Nunokawa, M., Saitoh, H., Srivastava, H.M., Close-to-convexity, starlikeness, and convexity of certain analytic functions, Appl. Math. Lett., 15(1)(2002), 63–69.
- Ponnusamy, S., Yamamoto, H., Yanagihara, H., Variability regions for certain families of harmonic univalent mappings, Complex Var. Elliptic Equ., 58(1)(2013), 23–34.
- Li, L., Ponnusamy, S., Disk of convexity of sections of univalent harmonic functions, J. Math. Anal. Appl., 408(2013), 589–596.
- Li, L., Ponnusamy, S., Injectivity of sections of univalent harmonic mappings, Nonlinear Anal., 89(2013), 276–283.
- Salagean, G.S., Subclass of univalent functions, in Complex Analysis-Fifth Romanian Finish Seminar, (1983), 362–372.
- Singh, R., Singh, S., Convolution properties of a class of starlike functions, Proc. Amer. Math. Soc., 106(1989), 145–152.
- Yalçın, S., Bayram, H., On harmonic univalent functions involving q-poisson distribution series, MJPS, 8(2)(2021).
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Reel ve Kompleks Fonksiyonlar |
| Bölüm | Makaleler |
| Yazarlar | |
| Yayımlanma Tarihi | 31 Aralık 2024 |
| Gönderilme Tarihi | 29 Haziran 2024 |
| Kabul Tarihi | 7 Kasım 2024 |
| Yayımlandığı Sayı | Yıl 2024 Cilt: 16 Sayı: 2 |