Document Type : Research Paper


Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan.


Let $f$ and $g$ be analytic in the open unit disc and, for $\alpha ,$ $\beta \geq 0$, let
J\left( \alpha ,\beta ,f,g\right) & =\frac{zf^{\prime }(z)}{f^{1-\alpha
}(z)g^{\alpha }(z)}+\beta \left( 1+\frac{zf^{\prime \prime }(z)}{f^{\prime
}(z)}\right) -\beta \left( 1-\alpha \right) \frac{zf^{\prime }(z)}{f(z)} \\
& \quad -\alpha \beta \frac{zg^{\prime }(z)}{g(z)}\text{.}
The main aim of this paper is to study the class of analytic functions which map $J\left( \alpha ,\beta ,f,g\right) $ onto conic regions. Several interesting problems such as arc length, inclusion relationship, rate of growth of coefficient and Growth rate of Hankel determinant will be discussed.


[1] D.A. Brannan, On functions of bounded boundary rotation, Proc. Edinburg Math. Soc., 16 (1969), pp. 339-347.

[2] G.M. Golusin, On distortion theorem and coefficients of univalent functions, Math. Sb., 19 (1946), pp. 183-203.

[3] A.W. Goodman, Univalent Functions, Vols. I & II, Polygonal Publishing House, Washington, New Jersey, (1983).

[4] W. K. Hayman, On functions with positive real part, J. London Math. Soc., 36 (1961), pp. 34-48.

[5] W.K. Hayman, On the second Hankel determinant of mean univalent functions, Proc. London Math. Soc., 18 (1968), pp. 77-84.

[6] S. Kanas and A. Wisniowska, Conic domain and starlike functions, Rev. Roumaine Math. Pures Appl., 45 (2000), pp. 647-657.

[7] S. Kanas and A. Wisniowska, Conic regions and k-uniform convexity, J. Comput. Math., 105 (1999), pp. 327-336.

[8] W. Kaplan, Close-to-convex Schlicht functions, Mich. Math. J., 1 (1952), pp. 169-185.

[9] S.S. Miller and P.T. Mocanu, Differential subordinations theory and applications, Marcel Dekker, Inc., New York, Basel, (2000).

[10] J.W. Noonan and D.K. Thomas, On the Hankel determinant of areally mean p-valent functions, Proc. London Math. Soc., 25 (1972), pp. 503-524.

[11] K.I. Noor, Hankel determinant problem for functions of bounded boundary rotations, Rev. Roum. Math. Pures Appl., 28 (1983), pp. 731-739.

[12] K.I. Noor, On a generalization of close-to-convexity, Int. J. Math. Math. Sci., 6 (1983), pp. 327-334.

[13] K.I. Noor, On the Hankel determinant of close-to-convex univalent functions, Inter. J. Math. Sci., 3 (1980), pp. 447-481.

[14] K.I. Noor, K. Ahmad, On higher order Bazilevic functions, Int. J. Mod. Phys. B, 27(2013), 14 pages.

[15] K.I. Noor, On the Hankel determinant problem for strongly close-to-convex functions, J. Natu. Geom., 11 (1997), pp. 29-34.

[16] K.I. Noor and Al-Naggar, Hankel determinant problem, J. Natu. Geom., 14 (1998), pp. 133-140.

[17] K.I. Noor and M.A. Noor, Higher order close-to-convex functions related with conic domains, Appl. Math. Inf. Sci., 8 (2014), pp. 2455-2463.

[18] K.S. Padmanabhan and R. Parvatham, Properties of a class of functions with bounded boundary rotation, Ann. Polon. Math., 31 (1975), pp. 311-323.

[19] B. Pinchuk, Functions with bounded boundary rotation, Israel J. Math., 10 (1971), pp. 7-16.

[20] Ch. Pommerenke, On starlike and close-to-convex functions, Proc. London Math. Soc., 13 (1963), pp. 290-304.

[21] D.K. Thomas, On Bazilevic functions, Trans. Amer. Math. Soc., 132 (1968), pp. 353-361.