Abstract
For normalized starlike functions f : 𝔻 → ℂ, we consider the analytic functions g : 𝔻 → ℂ defined by g(z) = (1 + z(f″(z))/f′(z))/(zf′(z)/f(z)) and g(z) = (1 − α)(zf′(z))/f(z) + α(1 + (zf″(z))/f′(z)), 0 ≤ α ≤ 1. We determine the largest radius ρ with 0 < ρ ≤ 1 such that g(ρ z) is subordinate to various functions with positive real part.
The first author is supported by Senior Research Fellowship from University Grants Commission, New Delhi
Communicated by Stanisława Kanas
References
[1] Ali, R. M.—Jain, N. K.—Ravichandran, V.: Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane, Appl. Math. Comput. 218(11) (2012), 6557–6565.10.1016/j.amc.2011.12.033Search in Google Scholar
[2] Arora, K.—Kumar, S. S.: Starlike functions associated with a petal shaped domain, Bull. Korean Math. Soc. 59(4) (2022), 993–1010.Search in Google Scholar
[3] Cho, N. E.—Kumar, V.—Kumar, S. S.— Ravichandran, V.: Radius problems for starlike functions associated with the sine function, Bull. Iranian Math. Soc. 45(1) (2019), 213–232.10.1007/s41980-018-0127-5Search in Google Scholar
[4] Gandhi, S.: Radius estimates for three leaf function and convex combination of starlike functions. In: Mathematical analysis. I. Approximation theory, Springer Proc. Math. Stat., 306, Springer, Singapore, 2018, pp. 173–184.10.1007/978-981-15-1153-0_15Search in Google Scholar
[5] Gandhi, S.—Ravichandran, V.: Starlike functions associated with a lune, Asian-Eur. J. Math. 10(4) (2017), Art. ID 1750064.10.1142/S1793557117500644Search in Google Scholar
[6] Gangania, K.: Theory of certain non-univalent analytic functions, Math. Slovaca 73(5) (2023), 1163–1182.10.1515/ms-2023-0086Search in Google Scholar
[7] Goel, P.—Kumar, S. S.: Certain class of starlike functions associated with modified sigmoid function, Bull. Malays. Math. Sci. Soc. 43(1) (2020), 957–991.10.1007/s40840-019-00784-ySearch in Google Scholar
[8] Janowski, W.: Some extremal problems for certain families of analytic functions. I, Ann. Polon. Math. 28 (1973), 297–326.10.4064/ap-28-3-297-326Search in Google Scholar
[9] Khatter, K.—Ravichandran, V.—Kumar, S. S.: Starlike functions associated with exponential function and the lemniscate of Bernoulli, Rev. R. Acad. Cienc. Exactas ís. Nat. Ser. A Mat. RACSAM 113(1) (2019), 233–253.10.1007/s13398-017-0466-8Search in Google Scholar
[10] Kumar, S. S.—Kamaljeet, G.: A cardioid domain and starlike functions, Anal. Math. Phys. 11(2) (2021), Art. No. 54.10.1007/s13324-021-00483-7Search in Google Scholar
[11] Kumar, S.—Ravichandran, V.: A subclass of starlike functions associated with a rational function, Southeast Asian Bull. Math. 40(2) (2016), 199–212.Search in Google Scholar
[12] Lecko, A.—Ravichandran, V.—Sebastian, A.: Starlikeness of certain non-univalent functions, Anal. Math. Phys. 11(4) (2021), Art. No. 163.10.1007/s13324-021-00600-6Search in Google Scholar
[13] Lee, S. K.—Khatter, K.—Ravichandran, V.: Radius of starlikeness for classes of analytic functions, Bull. Malays. Math. Sci. Soc. 43(6) (2020), 4469–4493.10.1007/s40840-020-01028-0Search in Google Scholar
[14] Ma, W. C.—Minda, D.: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis (Tianjin, 1992), Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA, 1992, pp. 157–169.Search in Google Scholar
[15] Madhumitha, S.—Ravichandran, V.: Radius of starlikeness of certain analytic functions, Rev. R. Acad. Cienc. Exactas ís. Nat. Ser. A Mat. RACSAM 115(4) (2021), Art. No. 184.10.1007/s13398-021-01130-3Search in Google Scholar
[16] Mendiratta, R.—Nagpal, S.—Ravichandran, V.: On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38(1) (2015), 365–386.10.1007/s40840-014-0026-8Search in Google Scholar
[17] Padmanabhan, K. S.: On certain classes of starlike functions in the unit disk, J. Indian Math. Soc. (N.S.) 32 (1968), 89–103.Search in Google Scholar
[18] Raina, R. K.—Sokół, J.: Some properties related to a certain class of starlike functions, C. R. Math. Acad. Sci. Paris 353(11) (2015), 973–978.10.1016/j.crma.2015.09.011Search in Google Scholar
[19] Ravichandran, V.—Rønning, F.—Shanmugam, T. N.: Radius of convexity and radius of starlikeness for some classes of analytic functions, Complex Variables Theory Appl. 33(1–4) (1997), 265–280.10.1080/17476939708815027Search in Google Scholar
[20] Ravichandran, V.—Sharma, K.: Sufficient conditions for starlikeness, J. Korean Math. Soc. 52(4) (2015), 727749.10.4134/JKMS.2015.52.4.727Search in Google Scholar
[21] Rønning, F.: Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118(1) (1993), 189–196.10.1090/S0002-9939-1993-1128729-7Search in Google Scholar
[22] Sebastian, A.—Ravichandran, V.: Radius of starlikeness of certain analytic functions, Math. Slovaca 71(1) (2021), 83–104.10.1515/ms-2017-0454Search in Google Scholar
[23] Shah, G. M.: On the univalence of some analytic functions, Pacific J. Math. 43 (1972), 239–250.10.2140/pjm.1972.43.239Search in Google Scholar
[24] Shanmugam, T. N.—Ramachandran, C.—Ravichandran, V.: Fekete-Szego problem for subclasses of starlike functions with respect to symmetric points, Bull. Korean Math. Soc. 43(3) (2006), 589–598.10.4134/BKMS.2006.43.3.589Search in Google Scholar
[25] Shanmugam, T. N.—Ravichandran, V.: Certain properties of uniformly convex functions, in Computational methods and function theory 1994 (Penang), 319–324, Ser. Approx. Decompos., 5, World Sci. Publ., River Edge, NJ.Search in Google Scholar
[26] Sharma, K.—Cho, N. E.—Ravichandran, V.: Sufficient conditions for strong starlikeness, Bull. Iranian Math. Soc. 47(5) (2021), 1453–1475.10.1007/s41980-020-00452-zSearch in Google Scholar
[27] Sharma, K.—Jain, N. K.—Ravichandran, V.: Starlike functions associated with a cardioid, Afr. Mat. 27(5–6) (2016), 923–939.10.1007/s13370-015-0387-7Search in Google Scholar
[28] Silverman, H.: Convex and starlike criteria, Int. J. Math. Math. Sci. 22(1) (1999), 75–79.10.1155/S0161171299220753Search in Google Scholar
[29] Singh, R.: On a class of star-like functions, Compositio Math. 19 (1967), 78–82.Search in Google Scholar
[30] Singh, V.: Remarks on a paper by H. Silverman, Int. J. Math. Math. Sci. 27(2) (2001), 65–68.10.1155/S0161171201010742Search in Google Scholar
[31] Sokół, J.—Stankiewicz, J.: Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. No. 19 (1996), 101–105.Search in Google Scholar
[32] Wani, L. A.—Swaminathan, A.: Radius problems for functions associated with a nephroid domain, Rev. R. Acad. Cienc. Exactas ís. Nat. Ser. A Mat. RACSAM 114(4) (2020), Art. No. 178.10.1007/s13398-020-00913-4Search in Google Scholar
© 2024 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- 10.1515/ms-2024-frontmatter4
- Intervals of posets of a zero-divisor graph
- Coalgebraic methods for Ramsey degrees of unary algebras
- On nonexistence of D(n)-quadruples
- Rees short exact sequences and preenvelopes
- Generalized discrete Grüss and related results with applications
- Radius problem associated with certain ratios and linear combinations of analytic functions
- Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions
- Oscillatory and asymptotic behavior of even-order nonlinear differential equations with mixed neutral terms
- On a solvable four-dimensional system of difference equations
- Euclidean operator radius inequalities of d-tuple operators and operator matrices
- Equable parallelograms on the Eisenstein lattice
- On certain star versions of the Hurewicz property using ideals
- Relative versions of star-Menger property
- The Maxwell-Boltzmann-Exponential distribution with regression model
- New results for the Marshall-Olkin family of distributions
- A new family of copulas based on probability generating functions
- Induced mappings on the hyperspace of totally disconnected sets
Articles in the same Issue
- 10.1515/ms-2024-frontmatter4
- Intervals of posets of a zero-divisor graph
- Coalgebraic methods for Ramsey degrees of unary algebras
- On nonexistence of D(n)-quadruples
- Rees short exact sequences and preenvelopes
- Generalized discrete Grüss and related results with applications
- Radius problem associated with certain ratios and linear combinations of analytic functions
- Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions
- Oscillatory and asymptotic behavior of even-order nonlinear differential equations with mixed neutral terms
- On a solvable four-dimensional system of difference equations
- Euclidean operator radius inequalities of d-tuple operators and operator matrices
- Equable parallelograms on the Eisenstein lattice
- On certain star versions of the Hurewicz property using ideals
- Relative versions of star-Menger property
- The Maxwell-Boltzmann-Exponential distribution with regression model
- New results for the Marshall-Olkin family of distributions
- A new family of copulas based on probability generating functions
- Induced mappings on the hyperspace of totally disconnected sets