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General Mathematics Vol. 17, No. 4 (2009), 205–210 On some integral operators on analytic functions M. Acu, A. Branga, D. Breaz, N. Breaz, E. Constantinescu, A. Totoi Abstract 2000 Mathematics Subject Classification: Primary 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 }. 205 206 M. Acu, A. Branga, D. Breaz, N. Breaz, E. Constantinescu, A. Totoi 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. [5] 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. [5] 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. [1] 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) On some integral operators on analytic functions 207 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. [1] 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 [2]) 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 [3] 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 T T0 (α, β) = S ∗ (α, β) T , where S ∗ (α, β) is the class defined by (4)). Definition 2. [3] 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} 208 M. Acu, A. Branga, D. Breaz, N. Breaz, E. Constantinescu, A. Totoi 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. [3] 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 From [4] we have the following definitions: Z ∞ P n an z , satisfies Vµ (f )(z) = Let f (z) ∈ T , f (z) = z − 1 f (tz) dt, t n=2 0 where µ is a real-valued, non-negative weight function normalized so that Z 1 µ(t)dt = 1. 0 µ ¶δ−1 (c + 1)δ c 1 If µ(t) = t log (c > −1 ; δ > 0), which gives the Koµ(δ) t matu operator. Then we have (8) Vµ (f )(z) = z − ¶δ ∞ µ X c+1 n=2 µ Remark 1. We notice that 0 < c+1 c+n c+n µ(t) an z n . ¶δ < 1, where c > −1 , δ > 0 and j ≥ 2. Remark 2. It is easy to prove, by using Theorem 1 and Remark 1, that for F (z) ∈ T and f (z) = Vµ (F )(z), we have f (z) ∈ T , where Vµ is the integral operator defined by (8). On some integral operators on analytic functions Theorem 5. Let F (z) be in the class T ∗ (α), α ∈ [0, 1), F (z) = z − 209 ∞ P aj z j , j=2 aj ≥ 0, j ≥ 2. Then f (z) = Vµ (F )(z) ∈ T ∗ (α), where Vµ is the integral operator defined by (8). Proof. From Remark 2 we obtain f (z) = Vµ (F )(z) ∈ T . ¶δ µ ∞ P c+1 j aj z j . By using Remark We have f (z) = z− bj z , where bj = c+j j=2 j−α j−α 1 we obtain bj < aj , for j = 2, 3, . . . , 0 ≤ α < 1, 1−α 1−α ∞ ∞ X X j−α j−α and thus bj ≤ aj ≤ 1. This mean (see Theorem 2) 1−α 1−α j=2 j=2 that f (z) = Vµ (F )(z) ∈ T ∗ (α). Similarly (by using Remark 2 and the 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) = Vµ (F )(z) ∈ T c (α), where Vµ 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) = Vµ (F )(z) ∈ C ∗ (α, β), F (z) = z − j=2 where Vµ 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) = Vµ (F )(z) ∈ Tn (α, β), where j=2 Vµ is the integral operator defined by (8). Remark 3. By choosing β = 1, respectively n = 0, in the above theorem, we obtain the similarly results for the classes Tn (α) and S ∗ (α, β). 210 M. Acu, A. Branga, D. Breaz, N. Breaz, E. Constantinescu, A. Totoi References [1] V.P. Gupta, P.K. Jain, Certain classes of univalent functions with negative coefficients, Bull. Austral. Math. Soc., 14(1976), 409-416. [2] G.S. Sălăgean, Subclasses of univalent functions, Complex Analysis. Fifth Roumanian-Finnish Seminar, Lectures Notes in Mathematics, 1013, Springer-Verlag, 1983, 362-372. [3] G.S. Sălăgean, Analytic functions with negative coefficients, Mathematica, 36(59), 2(1994), 219-224. [4] S. Shams et. al., Classes of Ruscheweyh-type analytic univalent functions, Southwest J. Pure Appl. Math., 2(2003), 105-110. [5] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51(1975), 109-116. M. Acu, A. Branga, E. Constantinescu, A. Totoi Department of Mathematics, ”Lucian Blaga” University from Sibiu 550012 Sibiu, Romania Corresponding e-mail: acu [email protected] Daniel Breaz, Nicoleta Breaz Department of Mathematics, ”1 Decembrie 1918” University of Alba Iulia 510009 Alba, Romania E-mails: [email protected], [email protected]