Generalizations of modified-Hadamard products of -valent functions with negative coefficients

✍ Scribed by H.E. Darwish; M.K. Aouf

• Book ID: 104046305
• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 433 KB
• Volume: 49
• Category: Article
• ISSN: 0895-7177
• DOI: 10.1016/j.mcm.2007.08.016

✦ Synopsis

Let T (n, p) denote the class of functions of the form f (z) = z p -∞ k=n a k+ p z k+ p (a k+ p ≥ 0; p, n ∈ N ) which are analytic and p-valent in the open unit disc U = {z : |z| < 1}. For functions f j (z) ( j = 1, 2) belonging to T (n, p), generalizations of the modified-Hadamard product of f 1 (z) and f 2 (z) represented by ( f 1 ∆ f 2 ) (r, s; z) (r, s ∈ R) are introduced. In this paper, we obtain several results of these generalizations of the modified-Hadamard products of functions in the classes T n ( p, α) and C n ( p, α). Also we obtain the order of p-valent starlikeness of a p-valent convex function of order α (0 ≤ α < p), with negative coefficients.