Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/hindawi-publishing-corporation/a-note-involving-p-valently-bazilevi-functions-PyJH6vxLbE
References
References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.
- Publisher
- Hindawi Publishing Corporation
- Copyright
- Copyright © 2005 Hindawi Publishing Corporation.
- ISSN
- 0161-1712
- eISSN
- 1687-0425
- Publisher site
- See Article on Publisher Site
Abstract
n (p) be the class of normalized functions of the form f (z) = z p + ∞ ak z k n, p ∈ N = {1,2,3,...} , (1.1) k=n+p which are analytic and p-valent in the unit disc = {z ∈ C : |z| < 1}. A function f ∈ n (p) is said to be in the class n (p,α) if it satisï¬es the inequality e z f (z) >α f (z) 0 ≤ α < p, p ∈ N, z ∈ . (1.2) c Also a function f ∈ n (p) is said to be a p-valently Bazilevi´ function of type ( ≥ 0) and order γ (0 ≤ γ < p; p ∈ N) if there exists a function g belonging to the class n (p) := n (p,0) such that e 1− >γ (z ∈ ). (1.3) We denote the class of all such functions by n (p,,γ). In particular, when = 1, a function f ∈ n (p,γ) := n (p,1,γ) is said to be p-valently close-to-convex of order γ in . Moreover, n (p,0,γ) =: n (p,γ) when = 0. 2. Main results and their consequences We begin with the following lemma due to Jack [2]. Copyright © 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:7 (2005) 1149–1153 DOI: 10.1155/IJMMS.2005.1149 A note involving p-valently Bazilevi´ functions c Lemma 2.1. Let ω(z) be nonconstant and regular in with ω(0) = 0. If |ω(z)| attains its maximum value on the circle |z| = r (0 < r < 1) at the point z0 , then z0 ω (z0 ) = cω(z0 ), where c ≥ 1. With the aid of the above lemma, we prove the following result. Theorem 2.2. Let f ∈ n (p), w ∈ C \ {0}, ≥ 0, 0 ≤ α < p, p ∈ N, z ∈ the function be deï¬ned by  , and also let (z) = ï£ z f (z) z f (z) − p f (z) If 1−   · 1 + z f (z) − (1 − ) z f (z) − zg (z) , f (z) f (z) (2.1) where g ∈ n (p). (z) satisï¬es one of the following conditions:  < |w |−2 e{w }    (z) = 0   > |w |−2 e{w } when when when e{w} > 0, e{w} = 0, e{w} < 0, (2.2) or  < |w |−2 m{w }    (z) = 0   > |w |−2 m{w } when m{w} > 0, when m{w} = 0, when m{w} < 0, (2.3) then  ï£ 1− w − p < p − α, (2.4) where the value of complex power in (2.4) is taken to be as its principal value. Proof. We deï¬ne the function Ω by  ï£ ï£¶w 1− − p = (p − α)Ω(z), (2.5) where ≥ 0, w ∈ C \ {0}, 0 ≤ α < p, p ∈ N, z ∈ , f ∈ n (p), and g ∈ n (p). We see clearly that the function Ω is regular in and Ω(0) = 0. Making use of the logarithmic differentiation of both sides of (2.5) with respect to the known complex variable z, and if we make use of equality (2.5) once again, then we ï¬nd that  wz ï£ 1−  −1  − p ï£ 1−  − p = zΩ (z) , Ω(z) (2.6) H¨ seyin Irmak et al. u which yields (z) := w zΩ (z) |w |2 Ω(z) w ∈ C \ {0 }; z ∈ (2.7) Assume that there exists a point z0 ∈ |z|≤|z0 | such that =1 max Ω(z) = Ω z0 (z ∈ ). (2.8) Applying Lemma 2.1, we can then write z0 Ω z0 = cΩ z0 Then (2.7) yields e so that e z0 (c ≥ 1). (2.9) z0 w z0 Ω z0 |w |2 Ω z0 e c w |w|−2 , (2.10) if c e{w} = 0 if |w |2   −2 ≤ |w | e{w} if  ≥ |w |−2 m{w }    m{w} = 0   ≤ |w |−2 m{w }  ≥ |w |−2 e{w }    e{w} > 0, e{w} = 0, e{w} < 0, (2.11) z0 c = |w |2 if m{w} > 0, if m{w} = 0, if m{w} < 0. (2.12) But the inequalities in (2.11) and (2.12) contradict, respectively, the inequalities in (2.2) and (2.3). Hence, we conclude that |Ω(z)| < 1 for all z ∈ . Consequently, it follows from (2.5) that 1− w −p = (p − α) Ω(z) < p − α. (2.13) Therefore, the desired proof is completed. This theorem has many interesting and important consequences in analytic function theory and geometric function theory. We give some of these with their corresponding geometric properties. First, if we choose the value of the parameter w as a real number with w := δ ∈ R \ {0} in the theorem, then we obtain the following corollary. Corollary 2.3. Let f ∈ n (p), δ ∈ R \ {0}, ≥ 0, 0 ≤ α < p, p ∈ N, z ∈ function be deï¬ned by (2.1). Also, if satisï¬es the following conditions: e (z)  > |δ |−2  < |δ |−2 , and let the when δ > 0, when δ < 0, (2.14) 1152 then A note involving p-valently Bazilevi´ functions c 1− > p − (p − α)1/δ . (2.15) Putting w = 1 in the theorem, we get the following corollary. Corollary 2.4. Let f ∈ n (p), g ∈ n (p), ≥ 0, 0 ≤ α < p, p ∈ N, z ∈ , and let the function be deï¬ned by (2.1). If (z) satisï¬es one of the following conditions: e then f ∈ n (p,,α), (z) < 1 or (z) = 0, (2.16) . that is, f is a p-valently Bazilevi´ function of type and order α in c n (p), 0 ≤ α < Setting w = 1 and = 0 in the theorem, we have the following corollary. Corollary 2.5. Let f ∈ by (z) = p, p ∈ N, z ∈ , and let the function be deï¬ned z f (z) z f (z) − p f (z) 1+ z f (z) z f (z) − . f (z) f (z) (2.17) If (z) satisï¬es one of the following conditions: e then f ∈ n (p,α), (z) < 1 or (z) = 0, . (2.18) that is, f is p-valently starlike of order α in g∈ By taking w = 1 and = 1 in the theorem, we obtain the following corollary. Corollary 2.6. Let f ∈ be deï¬ned by (z) = If n (p), n (p), 0 ≤ α < p, p ∈ N, z ∈ , and let the function z f (z) z f (z) − p 1+ z f (z) zg (z) − . f (z) (2.19) (z) satisï¬es one of the following conditions: e (z) < 1 or m (z) = 0, . (2.20) then f ∈ n (p,α), that is, f is p-valently close-to-convex of order α in Lastly, if we take p = 1 in Corollaries 2.4, 2.5, and 2.6, then we easily obtain the three important results involving Bazilevi´ functions of type ( ≥ 0) and order α (0 ≤ α < 1) c in , starlike functions of order α (0 ≤ α < 1) in , and close-to-convex functions of order α (0 ≤ α < 1) in , respectively, (see, e.g., [1, 3, 4, 5]). H¨ seyin Irmak et al. u Acknowledgments This work has been carried out by the help of four-month ï¬nancial support (June– ¨ Ë™ September, 2004) from the TUBITAK (The Scientiï¬c and Technical Research Council of Turkey) which is given to the ï¬rst author during the scientiï¬c research at the University ´ ´ of Rzeszow and Rzeszow University of Technology in Poland. This present investigation was also supported by NATO and Baskent University (Ankara, Turkey). The ï¬rst author ¸ would also like to acknowledge Professor Mehmet Haberal, Rector of Baskent University, who generously supports scientiï¬c researches in all aspects. I would like to extend my thanks to Professor J. Stankiewicz and Professor J. Dziok for their kind invitation to Poland and their invaluable support for this research. References [1] [2] [3] [4] [5] M. P. Ch’ˆ n and S. Owa, Notes on certain p-valently Bazilevi´ functions, Panamer. Math. J. 3 e c (1993), no. 4, 51–59. I. S. Jack, Functions starlike and convex of order α, J. London Math. Soc. (2) 3 (1971), 469–474. S. S. Miller, Distortions properties of alpha-starlike functions, Proc. Amer. Math. Soc. 38 (1973), 311–318. S. S. Miller, P. T. Mocanu, and M. O. Reade, All α-convex functions are univalent and starlike, Proc. Amer. Math. Soc. 37 (1973), 553–554. P. T. Mocanu, Une propri´t´ de convexit´ g´n´ralis´e dans la th´orie de la repr´sentation conforme, ee e e e e e e Mathematica (Cluj) 11 (34) (1969), 127–133 (French). H¨ seyin Irmak: Department of Mathematics Education, Faculty of Education, Baskent University, u ¸ Ba˘ lica Campus, 06530 Etimesgut, Ankara, Turkey g E-mail address: hisimya@baskent.edu.tr ´ Krzysztof Piejko: Department of Mathematics, Faculty of Management and Marketing, Rzeszow ´ University of Technology, 2 Wincentego Pola Street, 35-959 Rzeszow, Poland E-mail address: piejko@prz.rzeszow.pl ´ Jan Stankiewicz: Department of Mathematics, Faculty of Management and Marketing, Rzeszow ´ University of Technology, 2 Wincentego Pola Street, 35-959 Rzeszow, Poland; Institute of Mathemat´ ´ ics, University of Rzeszow, 16A Rejtana Street, 35-310 Rzeszow, Poland E-mail address: jstan@prz.rzeszow.pl
Journal
International Journal of Mathematics and Mathematical Sciences – Hindawi Publishing Corporation
Published: Jan 3, 2016