AREMARKONCERTAINDIFFE_省略_VINGp_VALENTFUN.pdfVIP

Anal. Theory Appl. Vol. 28, No. 1 (2012), 58–64 A REMARK ON CERTAIN DIFFERENTIAL INEQUALITIES INVOLVING p-VALENT FUNCTIONS Sukhwinder Singh Billing (Baba Banda Singh Bahadur Engineering College, India) Received Aug. 19, 2011 Abstract. In the present paper, we study certain differential inequalities involving p-valent functions and obtain sufficient conditions for uniformly p-valent starlikeness and uniformly p-valent convexity. We also offer a correct version of some known criteria for uniformly p-valent starlike and uniformly p-valent convex functions. Key words: p-valent function, uniformly starlike function, uniformly convex function, uni- formly close-to-convex function AMS (2010) subject classification: 30C80, 30C45 1 Introduction Let Ap denote the class of functions of the form f (z) = zp + ∞ ∑ k=p+1 akz k, p ∈ N = {1,2,3, · · · }, which are analytic and p-valent in the open unit disk E = {z ∈ C : |z| 1}. A function f ∈ Ap is said to be uniformly p-valent starlike in E if ? ( z f ′(z) f (z) ) ∣∣∣∣ z f ′(z) f (z) ? p ∣∣∣∣ , z ∈ E. (1.1) We denote byUS?p, the class of uniformly p-valent starlike functions. A function f ∈ Ap is said to be uniformly p-valent convex in E if ? ( 1+ z f ′′(z) f ′(z) ) ∣∣∣∣1+ z f ′′(z) f ′(z) ? p ∣∣∣∣ , z ∈ E. Let UCp denote the class of uniformly p-valent convex functions. A function f ∈ Ap is said to be uniformly p-valent close-to-convex in E if ? ( z f ′(z) g(z) ) ∣∣∣∣ z f ′(z) g(z) ? p ∣∣∣∣ , z ∈ E, (1.2) Anal. Theory Appl., Vol. 28, No.1 (2012) 59 for some g∈US?p. LetUCCp denote the class of all such functions. Note that the function g(z)≡ zp ∈Ap and satisfies the condition (1.1). Therefore, when we select g(z)≡ zp, in condition (1.2), it reduces to ? ( f ′(z) zp?1 ) ∣∣∣∣ f ′(z) zp?1 ? p ∣∣∣∣ , z ∈ E. (1.3) Hence, a function f ∈ Ap is uniformly p-valent close-to-convex in E if it satisfies the condition (1.3). In 1991, Goodman[2] introduced the concept of uniformly starlike and uniformly convex functions. He defined uniforml