On Univalent Functions Defined by the Multiplier Differential Operator

Transcription

1 Int. Journal of Math. Analysis, Vol. 6, 2012, no. 15, On Univalent Functions Defined by the Multiplier Differential Operator Li Zhou College of Mathematics and Information Science JiangXi Normal University NanChang , China Qing-hua Xu College of Mathematics and Information Science JiangXi Normal University NanChang , China Abstract The object of this paper is to obtain some properties of functions belonging to a new class, denoted by Rδ n (λ, α), using the multiplier differential operator. Keywords: Analytic functions, starlike functions, univalent functions, multipliter operator, Sǎlǎgean derivative operator 1 Introduction Let R = (, + ) be the set of real numbers, C be the set of complex numbers, be the set of positive integers, and N = {1, 2, 3,...} N 0 = N {0}.

2 736 Li Zhou and Qing-hua Xu Let A denote the family of functions of the form which are analytic in the open unit disc f() = + a j j, (1) j=2 U = { : C and < 1}. For a function f() ina, the multipliter operator D n α,δf() was extended by Deni and Orhan in [11] as follows: D 0 λ,δf() =f() D 1 λ,δ f() =D λ,δf() =λδ 2 f () +(λ δ)f () +(1 λ + δ)f(). D n λ,δ f() =D λ,δ(dλ,δ n 1 f()) where λ δ 0 and n N 0. If f Ais given by (1) then from the definition of the operator D n λ,δf(), it is easy verity that D n λ,δ f() = + φ n k a k k, where φ k =[1+(λδk + λ δ)(k 1)], (φ n k =[φ k] n ); λ δ 0 and n N 0. Remark 1. D n α,δf() is a generaliation of many other linear operators considered earlier. In particular, for f() ina we have the following : D n 1,0 f() Dn f() the operator defined by Sǎlǎgean (see [7]) D n λ,0f() D n λf() (see [8]) D n λ,δf() the same operator for 0 δ λ 1. 2 Preliminary Notes By using the multipliter differential operator Dα,δ n, we now define the following new subclass of functions belonging to the class A. Definition 2.1 Let Rδ n (λ, α) denote the class of functions f() Awhich satisfy the condition R((D n λ,δ f()) ) >α, U, for some 0 α 1, λ δ 0 and n N 0. Remark 2. It is clear that R 0 0(λ, α) R(α) R n 0 (0,α) and that R 1 0(λ, α) R 0 (λ, α), the class of functions f Asatisfying R(f ()+λf ()) >α, U,

3 Univalent functions 737 studied by Ponnusamy[4] and others. Definition 2.2 The Hadamad product or convolution of two power series f() = a k k and g() = b k k is defined as the power series (f g)() = k=0 k=0 a k b k k, U. k=0 The object of this paper is to derive several interesting properties of the class Rδ n (λ, α) such as inclusion relations, some convolution properties, and other results. 3 Main Results Theorem 3.5 shows that the functions in Rδ n (λ, α) belong to R(α) and hence are univalent. We need the following lemmas. Lemma 3.1 If p() is analytic in U and Rp() > 1/2, U, the function p*f takes its values in the convex hull of F (U). The assertion of Lemma 3.1 follows by using the Herglot representation for p. The next lemma is due to Fejér[2]. Definition 3.2 A sequence a 0,a 1,...,a n,...of nonnegative numbers is called a convex null sequence if a n 0 as n and a 0 a 1 a 1 a 2... a n a n Lemma 3.3 Let {c k } k=0 be a convex null sequence. Then the function p() = c 0 /2 + c k k, U, is analytic and Rp() > 0 in U. k=1 Lemma 3.4 Let q be a convex function in U and let h() =q()+nαq () where α>0 and n is a positive integer. If p H(U) with and then p() =q(0) + p n () n +... p()+αp () h()

4 738 Li Zhou and Qing-hua Xu and this result is sharp. p() q() Now we prove the following theorem. Theorem 3.5 R n+1 δ (λ, α) Rδ n (λ, α). Proof. Let f Rδ n+1 (λ, α) and let it be given by (1). Then we have R((Dλ,δ n+1 ) )=R(1 + k[1+(λδk + λ δ)(k 1)] n+1 a k k 1 ) >α, that is, Now R( (1 α) k[1+(λδk + λ δ)(k 1)] n+1 a k k 1 ) > 1/2 (Dλ,δf()) n =1+ k[1+(λδk + λ δ)(k 1)] n a k k 1 )= (1 + 1 k[1+(λδk + λ δ)(k 1)] n+1 a 2(1 α) k k 1 )) (1 + 2(1 α) k 1 ). 1+(λδk+λ δ)(k 1) Applying Lemma 3.3, with c 0 = 1 and c k = get R(1 + 2(1 δ) 1 [1+(λδk+λδ+λ δ)k], k=1,2,...,we k 1+(λδk+λ δ)(k 1) ) >α. Applying Lemma 3.1 to (D n λ,δ f()), we get the required result. Theorem 3.6 The set R n δ (λ, α) is convex. Proof. Let the functions f i () = + a ki k (i=1,2) be in the class Rδ n (λ, α). It is sufficient to show that the function h() = μ 1 f 1 ()+μ 2 f 2 (), with μ 1 and μ 2 nonnegative and μ 1 + μ 2 = 1, is in the class Rδ n (λ, α). Since h() = + (μ 1 a k1 + μ 2 a k1 ) k.

5 Univalent functions 739 then we have hence (Dλ,δh()) n =1+ k(μ 1 a k1 + μ 2 a k1 )[1 + (λδ + λ δ)(k 1)] n k 1 ; R(Dλ,δ n h()) = R(1 + μ 1 k[1+(λδk + λ δ)(k 1)] n a k1 k 1 )+ R(1 + μ 2 k[1+(λδk + λ δ)(k 1)] n a k2 k 1 ) 1 (2) Since f 1, f 2 Rδ n (λ, α), this implies that R(1 + μ i k[1+(λδk + λ δ)(k 1)] n a ki k 1 ) > 1+μ i (α 1) (3) Using (3) in (2), we obtain R(D n λ,δ h()) > 1+α(μ 1 + μ 2 ) (μ 1 + μ 2 ) and since (μ 1 + μ 2 ) = 1, the theorem is proved. Hallenbeck[3] showed that Rf () >α R f() > (2α 1) + 2(1 α)log2. (4) Using Theorem 3.5 and (4), we obtain the following theorem. Theorem 3.7 Let f Rδ n (λ, α), then R Dn λ,δ f() > (2α 1) + 2(1 α)log2. Theorem 3.8 Let q be a convex function with q(0)=1 and let h be a function of the form h() =q()+q (), λ > 0, U. If f Averifies the differential subordination (D n λ,δ f()) h(), U (5) then and this result is sharp. D n λ,δf() q() (6)

6 740 Li Zhou and Qing-hua Xu Proof. If we let then we obtain The subordination(5) becomes p() = Dn λ,δ f(), U (D n λ,δ f()) = p()+p (), U. p()+p () q()+q () and from Lemma3.4 we have (6). The result is sharp. Remark 3. For λ =1 this result was obtained in[10]. Ruscheweyh and Sheil-Small [5] verified the Polya-Schoenberg conjecture and its analogous results, that is, C C C, C S S and C K K, where C, S and K denote the classes of convex, starlike, and close-to-convex univalent functions, respectively. Next, we prove the analogue of the Polya- Schoenberg conjecture for the class R n δ (λ, α). Theorem 3.9 Let f R n δ (λ, α) and g C. Then f*g R n δ (λ, α) Proof. It is known that if g is convex univalent in U, then R g() > 1/2. Using convolution properties, we have R(Dλ,α n (f g)()) = R((Dλ,α n f()) g() ) (7) and the result follows by application of Lemma 3.1. Theorem 3.10 Let f and g Rδ n(λ, α), then f*g Rn δ (λ, β), where Proof. Let g() = β = λ(2α+1)+4α 1 2(λ+1) α. b k k Rδ n (λ, α), then R(1 + k[1+(λδk + λ δ)(k 1)] n b k k 1 ) >α. (8) Let c 0 = 1 and c k = λ+1 (k+1)[1+(λδk+λδ+λ δ)k] n, k 1.

7 Univalent functions 741 Then {c k } k=0 is convex null sequence. Hence, by Lemma 3.3, we have R(1 + λ +1 k[1+(λδk + λ δ)(k 1)] n k 1 ) > 1/2. (9) Now we take the convolution of (4) and (5) and apply Lemma 3.1 to obtain or Hence R(1 + (λ +1) b k k 1 ) >α R g() = R(1 + R( g() b k k 1 ) > λ+α λ+1. 2α+λ 1 2(λ+1) ) > 1/2. Since f R n δ (λ, α), by applying Lemma3.1, we obtain R((D n λ,δ f()) ( g() 2α+λ 1 )) >α 2(λ+1) or R((Dλ,δf()) n g() ) > λ(2α+1)+4α 1 = β, 2(λ+1) and by(7), the result follows. Remark 4. If we set λ = 0 and δ = 0 in Theorem 3.10, we get the result for functions in R(α), given by Ahuja [1]. References [1] O. P. Ahuja, nivalent functions whose derivatives have a positive real part, Rend.Mat, 2 (1982), [2] L. Fejér, Ü ber, die positivität von summen, die nach trigonometrischen order Legendreschen funktionen fortschreiten, I, Acta Seged, 2(1925), [3] D. J. Hallenbeck, Convex hulls and extreme points of some families of univalent functions, Trans. Amer. Math. Soc, 192 Wiley, (1947), [4] S. Ponnusamy, Differential subordination and starlike functions, Complex Variable Theory Appl, 19 (1992),

8 742 Li Zhou and Qing-hua Xu [5] St. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture, Comment. Math. Helv, 48 (1973), [6] H. Saitoh, Properties of certain analytic functions, Proc. Japan Acard. Ser. A Math. Sci, 65 (1989), [7] G. S. Sǎlǎgean, Subclasses of univalent functions, Complex Analysis-Fifth Romanian-Finnish Seminar, 1013 (1983), [8] Al-Oboudi. F.M, On univalent functions defined by a generalied Sǎlǎgean operator, Inter. J. of Math. and Mathematical Sci, 27 (2004), [9] Miller, S.S, Mocanu, P.T, On some classes of first order differential subordinations, Michigan Math. J,32 (1985), [10] Oros. G. I, A class of holomorphic functions defined using a differential operator, General Mathematics Sibiu, [11] M. Çağlar, E. Deni, H. Orhan, Coefficient bounds for a subclass of starlike functions of complex o0rder, Appl. Math and Comput,2 (1982), Received: October, 2011