Hankel Determinant for -Valent Alpha-Convex Functions
The objective of the present paper is to obtain the sharp upper bound of for p-valent α-convex functions of the form in the unit disc .
Let be the class of analytic functions of the form in the unit disc with . Let be the subclass of , consisting of univalent functions.
is the class consisting of functions of the form (1) and satisfying the condition
The functions of the class are called p-valent starlike functions. In particular, , the class of starlike functions.
is the class of functions of the form (1), satisfying the condition
The functions of the class are known as p-valent convex functions. Particularly, , the class of convex functions.
Obviously if and only if .
Let be the class of functions of the form (1), satisfying the condition
Functions in the class are known as -valent alpha-convex functions. For , the class reduces to the class of alpha-convex functions introduced by Mocanu . Also and .
In 1976, Noonan and Thomas  stated the th Hankel determinant for and as
This determinant has also been considered by several authors. For example, Noor  determined the rate of growth of as for functions given by (1) with bounded boundary. Ehrenborg  studied the Hankel determinant of exponential polynomials. Also Hankel determinant was studied by various authors including Hayman  and Pommerenke . In , Janteng et al. studied the Hankel determinant for the classes of starlike and convex functions. Again Janteng et al. discussed the Hankel determinant problem for the classes of starlike functions with respect to symmetric points and convex functions with respect to symmetric points in  and for the functions whose derivative has a positive real part in . Also Hankel determinant for various subclasses of -valent functions was investigated by various authors including Krishna and Ramreddy  and Hayami and Owa .
Easily, one can observe that the Fekete and Szegö functional is . Fekete and Szegö  then further generalised the estimate , where is real and . For our discussion in this paper, we consider the Hankel determinant in the case of and :
In this paper, we seek sharp upper bound of the functional for functions belonging to the class . The results due to Janteng et al.  follow as special cases.
2. Preliminary Results
Let be the family of all functions analytic in for which and for .
Lemma 1 (see ). If , then
3. Main Result
Theorem 3. If , then where
Proof . Since , so from (4)
On expanding and equating the coefficients of , and in (11), we get
Equation (12) yields:
Using Lemmas 1 and 2 in (13), we obtain Assume that and ; using triangular inequality and , we have It is easy to verify that is an increasing function and so .
Consequently where So where is defined in (10).
Now gives is negative at .
So Hence from (15), we obtain (9).
The result is sharp for , , and .
For , Theorem 3 gives the following result.
Corollary 4. If , then
For , Theorem 3 yields.
Corollary 5. If , then
Corollary 6. If , then
Corollary 7. If , then
W. K. Hayman, Multivalent Functions, vol. 48 of Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, Cambridge, UK, 1958.View at: MathSciNet
Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.View at: MathSciNet
A. Janteng, S. A. Halim, and M. Darus, “Hankel determinant for functions starlike and convex with respect to symmetric points,” Journal of Quality Measurement and Analysis, vol. 2, no. 1, pp. 37–43, 2006.View at: Google Scholar
D. V. Krishna and T. Ramreddy, “Hankel determinant for p-valent starlike and convex functions of order α,” Novi Sad Journal of Mathematics, vol. 42, no. 2, pp. 89–96, 2012.View at: Google Scholar