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Abstract
A certain class Mk(a, (3) of analytic functions is introduced and it is shown that
M2(a,fJ) is contained in the class of Bazilcvic functions. Some other properties of M1,, 佤fJ) are
also derived.
Results & Lemmas (7)
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Lemma 2.1.
Lemma 2.1. Let p be analytic in E and p(O) (p+a严E P implies P in E. p k PE k The proof follows directly from the result, proved in (1),…
Lemma 2.1. Let p be analytic in E and p(O) (p+a严E P implies P in E. p k PE k The proof follows directly from the result, proved in (1), that functions with bounded Mocanu variation are in Rk. From the Herglotz representation (1.1) for k = 2, we have the following result 1. Then, for a~0, z E E,
Lemma 2.2.
Lemma 2.2. If p is analytic in E, p(O) = l and Rep(z) > !, z E E, then for any function F, analytic in E, the function p * F takes values…
Lemma 2.2. If p is analytic in E, p(O) = l and Rep(z) > !, z E E, then for any function F, analytic in E, the function p * F takes values in the convex hull of the image of E under F. 3. Main Results
Theorem 3.1.
Theorem 3.1. For a 2: 0, Mk(a, 3) C Mk(O, 3).
Theorem 3.1. For a 2: 0, Mk(a, {3) C Mk(O, {3).
Corollary 3.1. · radius
Corollary 3.1. Let g E S * and k = 2. Then, for a 2:: 0, M瓦/3) CS since in 珈.<;Case z!'(z)·EP,gES*irnpliesf f 丘f3 (z) 評(z) is Bazilevic and…
Corollary 3.1. Let g E S * and k = 2. Then, for a 2:: 0, M瓦/3) CS since in 珈.<;Case z!'(z)·EP,gES*irnpliesf f 丘f3 (z) 評(z) is Bazilevic and hence univalent. Corol區y 3.2. For o: ~0, Mk(a,O) C Mk(O,O). That is a function with bounded Mocanu variation is of bounded radius rotation. W聶k = 2 we also deduce that a-starlike functions are starlike.
Corollary 3.3.
Corollary 3.3. Let g ES* and k = 2, /3 = l. Then M瓦1) C M2(0, l) = K C S That is f E M2(a, 1) is a close-to-convex univalent function. In…
Corollary 3.3. Let g ES* and k = 2, /3 = l. Then M瓦1) C M2(0, l) = K C S That is f E M2(a, 1) is a close-to-convex univalent function. In the opposite direction we prove the following.
Theorem 3.2.
Theorem 3.2. Let f E Mk(0,(3). Then f E Mk(a,(3) for lzl < r。 where 1 Ta= 2a + v4a2 - 2a + 1 (3.2)
Theorem 3.2. Let f E Mk(0,(3). Then f E Mk(a,(3) for lzl < r。 where 1 Ta= 2a + v4a2 - 2a + 1 (3.2)
Corollary 3.5.
Corollary 3.5. In Corollary 3A we take a = 1. Then it follows that J E Rk implie.s f E Vk for·Iz I < —— =2- 翥and k = 2 gives us the radiu.s…
Corollary 3.5. In Corollary 3A we take a = 1. Then it follows that J E Rk implie.s f E Vk for·Iz I < —— =2- 翥and k = 2 gives us the radiu.s of conv.exity for starlike 2+ 矗 functions. References [1] H. B. Coonce and M. R. Ziegler, "Functions with bounded Mocanu variation," Rev. Roum. Math. Pures et Appl., 19 {1974), 1093-1104. [2] K. I. Noor, "On a generalization of close-to-convexity," Int. J. Math. and Math. Sci., 6 (1983), 327-334. 圍B. Pinchuk, "Functions with bounded boundary rotation," I. J.
Definitions (1)
Def 1.1.
Definition 1.1. A function f E A is said to be belong to the class Mk(o:, /3') if and only if it satisties the property J(o:, f3; f(z),…
Definition 1.1. A function f E A is said to be belong to the class Mk(o:, /3') if and only if it satisties the property J(o:, f3; f(z), g(z)) = { fl-『;# {z) +o: (1十氕訂-o:(1-{3)誓－磷了~~;)}E 1~和 for some real o:, /3(/j 乏0), g EA and z E E. Special Cases (i) Fork= 2, o: = O and g ES*, we obtain the sell-known class B(/3) CS of Bazilevic functions of type /3. (ii) When f3 = O, the class Mk(o:, 0) consists of functions with bounded Mocanu varia- tion, sec [1] and M2(o:, 0) is the class of o:-starlike fu
Function classes studied:
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