Starlikeness of integral transforms and duality. (English) Zbl 1244.30008

Let \(D\) be the complex unit disc and \(\mathcal A\) the class of functions \(f(z)=z+a_2z^2+\cdots\). For \(f\in\mathcal{A}\) also satisfing the condition \[ \mathrm{Re} e^{\mathrm{i}\phi}\left((1-\alpha+2\gamma)\frac{f(z)}{z}+(\alpha-2\gamma)f'(z)+\gamma zf'' (z)-\beta\right)> 0 \] for suitable \(\phi\), \(\alpha\), \(\beta\) and \(\gamma\), the authors give sufficient conditions so that the function defined by \[ V_{\lambda}(f)(z)=\int_0^z\lambda(t)\frac{f(tz)}{t}dt \] (with \(\lambda\) chosen so that the above formula generalizes some results of other authors, but also provide new results) is starlike. Particular cases of \(\lambda\) are taken into account. Some consequences are also given. One of them gives a sharp estimate for the real constant \(\beta<1\) that ensures starlikeness of a function \(f\in\mathcal A\) that satisfies the condition \(\mathrm{Re}(f'(z)+\alpha zf'' (z)+\gamma z^2f'''(z))>\beta\).


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
Full Text: DOI


[1] Ali, R.M.; Lee, S.K.; Subramanian, K.G.; Swaminathan, A., A third-order differential equation and starlikeness of a double integral operator, Abstr. appl. anal., (2011), Art. ID 901235, 10 pp · Zbl 1207.30012
[2] Balasubramanian, R.; Ponnusamy, S.; Vuorinen, M., On hypergeometric functions and function spaces, J. comput. appl. math., 139, 2, 299-322, (2002) · Zbl 1172.33302
[3] Balasubramanian, R.; Ponnusamy, S.; Prabhakaran, D.J., Duality techniques for certain integral transforms to be starlike, J. math. anal. appl., 293, 1, 355-373, (2004) · Zbl 1061.30012
[4] Balasubramanian, R.; Ponnusamy, S.; Prabhakaran, D.J., On extremal problems related to integral transforms of a class of analytic functions, J. math. anal. appl., 336, 1, 542-555, (2007) · Zbl 1125.30005
[5] Carlson, B.C.; Shaffer, D.B., Starlike and prestarlike hypergeometric functions, SIAM J. math. anal., 15, 4, 737-745, (1984) · Zbl 0567.30009
[6] Fournier, R.; Ruscheweyh, S., On two extremal problems related to univalent functions, Rocky mountain J. math., 24, 2, 529-538, (1994) · Zbl 0818.30013
[7] Hohlov, Yu.E., Convolution operators that preserve univalent functions, Ukrainian math. J., 37, 2, 220-226, (1985), 271 (in Russian)
[8] Kim, Y.C.; Rønning, F., Integral transforms of certain subclasses of analytic functions, J. math. anal. appl., 258, 2, 466-489, (2001) · Zbl 0982.44001
[9] Komatu, Y., On analytic prolongation of a family of operators, Math. (cluj), 32(55), 2, 141-145, (1990) · Zbl 0753.30005
[10] Ponnusamy, S., Differential subordinations concerning starlike functions, Proc. Indian acad. sci. math. sci., 104, 2, 397-411, (1994) · Zbl 0808.30012
[11] Ponnusamy, S.; Rønning, F., Duality for Hadamard products applied to certain integral transforms, Complex var. theory appl., 32, 3, 263-287, (1997) · Zbl 0878.30007
[12] Ponnusamy, S.; Rønning, F., Integral transforms of a class of analytic functions, Complex var. elliptic equ., 53, 5, 423-434, (2008) · Zbl 1146.30009
[13] Singh, R.; Singh, S., Convolution properties of a class of starlike functions, Proc. amer. math. soc., 106, 1, 145-152, (1989) · Zbl 0672.30007
[14] Ruscheweyh, S., Duality for Hadamard products with applications to extremal problems for functions regular in the unit disc, Trans. amer. math. soc., 210, 63-74, (1975) · Zbl 0311.30011
[15] Ruscheweyh, S., Convolutions in geometric function theory, Sem. math. sup., vol. 83, (1982), Presses Univ. Montréal Montreal, QC · Zbl 0499.30001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.