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General Mathematics Vol. 15, Nr. 2–3(2007), 184-189 Some preserving properties of a integral operator Ana-Maria Acu and Eugen Constantinescu Abstract In this paper we will give some preserving properties of some subclasses of functions with negative coefficients by using a integral operator. 2000 Mathematical Subject Classification: 30C45 Key words and phrases: Function with negative coefficients, integral operator, Sălăgean operator 1 Introduction and Preliminaries Let H(U ) be the set of functions which are regular in the unit disc U , A = {f ∈ H(U ) : f (0) = f 0 (0) − 1 = 0}, Hu (U ) = {f ∈ H(U ) : f is univalent in U } and S = {f ∈ A : f is univalent in U }. 184 Some preserving properties of a integral operator 185 We denote with T the subset of the functions f ∈ S, which have the form (1) f (z) = z − T ∞ X aj z j , aj ≥ 0 , j ≥ 2 , z ∈ U j=2 and with T ∗ = T S ∗ , T ∗ (α) = T T T S c (α), where 0 ≤ α < 1 . T S ∗ (α), T c = T T S c and T c (α) = Theorem 1. [6] For a function f having the form (1) the following assertions are equivalents: ∞ X (i) jaj ≤ 1 j=2 (ii) f ∈ T (iii) f ∈ T ∗ . Regarding the classes T ∗ (α) and T c (α) with 0 ≤ α < 1 , we recall here the following result: Theorem 2. [6] A function f having the form (1) is in the class T ∗ (α) if and only if: (2) ∞ X j−α j=2 1−α aj ≤ 1 , and is in the class T c (α) if and only if: (3) ∞ X j(j − α) j=2 1−α aj ≤ 1 . Definition 1. [2] Let S ∗ (α, β) denote the class of functions having the form (1) which are starlike and satisfy ¯ ¯ zf 0 (z) ¯ ¯ − 1 f (z) ¯ ¯ (4) ¯ zf 0 (z) ¯<β ¯ + (1 − 2α) ¯ f (z) 186 Ana-Maria Acu and Eugen Constantinescu for 0 ≤ α < 1 and 0 < β ≤ 1. And let C ∗ (α, β) denote the class of functions such that zf 0 (z) is in the class S ∗ (α, β). Theorem 3. [2] A function f having the form (1) is in the class S ∗ (α, β) if and only if: (5) ∞ X {(j − 1) + β(j + 1 − 2α)} aj ≤ 2β(1 − α) , j=2 and is in the class C ∗ (α, β) if and only if: (6) ∞ X j {(j − 1) + β(j + 1 − 2α)} aj ≤ 2β(1 − α) . j=2 Let Dn be the Sălăgean differential operator (see [3]) defined as: Dn : A → A , n ∈ N and D0 f (z) = f (z) D1 f (z) = Df (z) = zf 0 (z) , Dn f (z) = D(Dn−1 f (z)). In [5] the author define the class Tn (α, β), from which by choosing different values for the parameters we obtain variously subclasses of analytic functions with negative coefficients (for example Tn (α, 1) = Tn (α) which is the class of n-starlike of order α functions with negative coefficients and T0 (α, β) = S ∗ (α, β) is the class defined by (4)). Definition 2. [5] Let α ∈ [0, 1), β ∈ (0, 1] and n ∈ N . We define the class Sn (α, β) of the n-starlike of order α and type β through Sn (α, β) = {f ∈ A ; |J(f, n, α; z)| < β} Some preserving properties of a integral operator 187 Dn+1 f (z) − Dn f (z) where J(f, n, α; z) = n+1 , z ∈ U . Consequently n T D f (z) + (1 − 2α)D f (z) Tn (α, β) = Sn (α, β) T . Theorem 4. [5] Let f be a function having the form (1). Then f ∈ Tn (α, β) if and only if ∞ X (7) j n [j − 1 + β(j + 1 − 2α)] aj ≤ 2β(1 − α) . j=2 2 Main results In [1] the authors consider the integral operator Ic+δ : A → A, 0 < u ≤ 1, 1 ≤ δ < ∞, 0 < c < ∞, defined by Z1 (8) uc+δ−2 F (uz)du. f (z) = Ic+δ (F (z)) = (c + δ) 0 Remark 1. For F (z) = z + ∞ P aj z j , from (8) we obtain j=2 f (z) = z + ∞ X j=2 Also, we notice that 0 < c+δ aj z j . c+j+δ−1 c+δ < 1, where 0 < c < ∞, j ≥ 2, c+j+δ−1 1 ≤ δ < ∞. Remark 2. It is easy to prove that for F (z) ∈ T and f (z) = Ic+δ (F (z)), we have f (z) ∈ T , where Ic+δ is the integral operator defined by (8). 188 Ana-Maria Acu and Eugen Constantinescu Theorem 5. Let F (z) be in the class T ∗ (α), α ∈ [0, 1), F (z) = z − ∞ P aj z j , j=2 aj ≥ 0, j ≥ 2. Then f (z) = Ic+δ (F (z)) ∈ T ∗ (α), where Ic+δ is the integral operator defined by (8). Proof. From Remark 2 we obtain f (z) = Ic+δ (F (z)) ∈ T . ∞ X ∞ P c+δ j−α We have aj ≤ 1 and f (z) = z− bj z j , where bj = aj . 1−α c+j+δ−1 j=2 j=2 c+δ By using the fact that 0 < < 1, where 0 < c < ∞, j ≥ 2, c+j+δ−1 ∞ X j−α j−α j−α 1 ≤ δ < ∞, we obtain bj < aj and thus bj ≤ 1. This 1−α 1−α 1−α j=2 mean (see Theorem 2) that f (z) = Ic+δ (F (z)) ∈ T ∗ (α). Similarly (by using Theorems 2 , 3 and 4) we obtain: Theorem 6. Let F (z) be in the class T c (α), α ∈ [0, 1), F (z) = z − ∞ P aj z j , j=2 aj ≥ 0, j ≥ 2. Then f (z) = Ic+δ (F (z)) ∈ T c (α), where Ic+δ is the integral operator defined by (8). Theorem 7. Let F (z) be in the class C ∗ (α, β), α ∈ [0, 1), β ∈ (0, 1], ∞ P aj z j , aj ≥ 0, j ≥ 2. Then f (z) = Ic+δ (F (z)) ∈ C ∗ (α, β), F (z) = z − j=2 where Ic+δ is the integral operator defined by (8). Theorem 8. Let F (z) be in the class Tn (α, β), α ∈ [0, 1), β ∈ (0, 1], n ∈ N, ∞ P F (z) = z − aj z j , aj ≥ 0, j ≥ 2. Then f (z) = Ic+δ (F (z)) ∈ Tn (α, β), j=2 where Ic+δ is the integral operator defined by (8). Remark 3. By choosing β = 1, respectively n = 0, in the above theorem, we obtain the similarly result for the classes Tn (α) and S ∗ (α, β). Some preserving properties of a integral operator 189 References [1] M. Acu and all, On some starlike functions with negative coefficients, (to appear). [2] V.P. Gupta, P.K. Jain, Certain classes of univalent functions with negative coefficients, Bull. Austral. Math. Soc., 14(1976), 409-416. [3] G.S. Sălăgean, Geometria planului complex, Ed. Promedia Plus, ClujNapoca, 1997. [4] G.S. Sălăgean, Subclasses of univalent functions, Complex Analysis. Fifth Roumanian-Finnish Seminar, Lectures Notes in Mathematics, 1013, Springer-Verlag, 1983, 362-372. [5] G.S. Sălăgean, Analytic functions with negative coefficients, Mathematica, 36(59), 2(1994), 219-224. [6] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51(1975), 109-116. Department of Mathematics Faculty of Science University ”Lucian Blaga” of Sibiu Str. I. Ratiu, no. 5-7, 550012 - Sibiu, Romania Email addresses: [email protected] , [email protected]