On α-uniformly close-to-convex and. quasi-convex functions with. negative coefficients

Transcription

1 heoretical Mathematics & Applications, vol.3, no.2, 2013, ISSN: (print), (online) Scienpress Ltd, 2013 On α-uniformly close-to-convex and quasi-convex functions with negative coefficients Irina Dorca 1 Abstract In this paper we study a class of α-uniformly starlike functions with negative coefficients, a class of α-uniformly convex functions with negative coefficients, a class of α-uniformly close-to-convex functions with negative coefficients and a class of quasi-convex functions with negative coefficients. Mathematics Subject Classification: 30C45 Keywords: α-uniformly starlike functions, α-uniformly convex functions, α- uniformly close-to-convex functions, quasi-convex functions, negative coefficients. 1 Introduction Let H(U) be the set of functions which are regular in the unit disc U, A = {f H(U) : f(0) = f (0) 1 = 0} (1) 1 University of Piteşti. Romania. Article Info: Received : April 22, Revised : May 30, 2013 Published online : June 25, 2013

2 30 On α-uniformly close-to-convex and quasi-convex functions and S = {f A : f is univalent in U}. In [3] the subfamily of S consisting of functions f of the form f(z) = z was introduced. a j z j, a j 0, j = 2, 3,..., z U (2) Let (n, p) denote the class of functions of the form f(z) = z p a p + pz l+p,, a l+p 0, p, j N = {1, 2,...}, (3) p=j which are analytic in U. We have (1, 1) =. he purpose of this paper is to define a class of α-uniformly close-to-convex and quasi-convex functions with negative coefficients. For this, we make use of the following well known results, which are taken from literature. 2 Preliminary Results We begin with the assertions concerning the starlike functions with negative coefficients (e.g. heorem 2.1), we continue with the operator I c+δ (see (4)) and we end by recalling some known results from [5] and [6] that we use forward in our study. he methods used to prove our results are taken from literature. heorem 2.1. [2] If f(z) = z a j z j, a j 0, j = 2, 3,..., z U then the next assertions are equivalent: (i) ja j 1 (ii) f (iii) f, where = S and S is the well-known class of starlike functions. Definition 2.1. [2] Let α [0, 1) and n N, then { } S n (α) = f A : Re Dn+1 f(z) D n > α, z U f(z)

3 Irina Dorca 31 is the set of n-starlike functions of order α. Also, we denote n (α) = S n (α). In [1] is defined the integral operator: I c+δ : A A, c < u 1, 1 δ <, 0 < c <, with 1 f(z) = I c+δ (F (z)) = () 0 u c+δ 2 F (uz)du. (4) Remark 2.1. If F (z) = z + c j z j, the Also we notice that 0 < δ [1, ). f(z) = I c+δ (F (z)) = z + c + j + δ 1 c + j + δ 1 a jz j. < 1, where c (0, ), j 2, Remark 2.2. It is easy to prove that for F (z) and f(z) = I c+δ (F (z)) we have f(z), where I c+δ is the integral operator defined by (4). In [5] are presented the following classes of analytic functions: Definition 2.2. [5] following inequality: Let C S denote the class of functions in S satisfying the { } (zf (z)) Re > 0, (z U). (5) f (z) + f ( z) Definition 2.3. [5] Let US (k) (α, β) denote the class of functions in satisfying the following inequality: { } zf (z) Re > α zf (z) f k (z) f k (z) 1 + β, (z U), (6) where α 0, 0 β < 1, k 1 is a fixed positive integer and f k (z) are defined by the following equality: f k (z) = 1 k 1 ε ν f(ε ν z), (ε k = 1, z U). (7) k ν=0

4 32 On α-uniformly close-to-convex and quasi-convex functions If k = 1, then the class US (k) (α, β) reduces to the class of α-uniformly starlike functions of order β. If k = 2, α = 0 and β = 0, then the class US (k) (α, β) reduces to the class SS of starlike functions with respect to symmetric points. From [4] we know that if f(z) S, { } zf (z) Re > 0, z U. (8) f(z) f( z) Definition 2.4. [5] Let UCV (k) (α, β) denote the class of functions in satisfying the following inequality: { } (zf (z)) Re f k (z) > α (zf (z)) f k (z) 1 + β, (z U), (9) where α 0, 0 β < 1, k 1 is a fixed positive integer and f k (z) are defined by (7). If k = 1, then the class UCV (k) (α, β) reduces to the class of α-uniformly convex functions of order β. If k = 2, α = 0 and β = 0, then the class UCV (k) (α, β) reduces to the class CS. heorem 2.2. [5] Let α 0, 0 β < 1, k 1 be a fixed positive integer and f(z). hen f(z) US (k) (α, β) iff [(1 + α)(jk + 1) (α + β)] a jk+1 + (10) (1 + α)ja j < 1 β., j lk+1 heorem 2.3. [6] Let α 0, 0 β < 1, k 1 be a fixed positive integer and f(z). hen f(z) UCV (k) (α, β) if and only if (jk + 1)[(1 + α)(jk + 1) (α + β)] a jk+1 + (11) (1 + α)j 2 a j < 1 β., j lk+1

5 Irina Dorca 33 Definition 2.5. [6] Let C (k) (λ, α) denote the class of functions in A satisfying the following inequality: { } zf (z) + λz 2 f (z) Re (1 λ)f k (z) + λzf k (z) > α, (z U), (12) where 0 α < 1, 0 λ 1, k 2 is a fixed positive integer and f k (z) is defined by equality (7). Definition 2.6. [6] Let QC (k) (λ, α) denote the class of functions in A satisfying the following inequality: { } Re z λz2 f (z) + (2λ + 1)zf (z) + f (z) λz 2 f k (z) + zf k (z) > α, (z U), (13) where 0 α < 1, 0 λ 1, k 2 is a fixed positive integer and f k (z) is defined by equality (7). For convenience we write C (k) (λ, α) as C (k) (λ, α) and QC(k) (λ, α) as QC (k) (λ, α). heorem 2.4. [6] Let 0 α < 1, 0 λ < 1, k 2 be a fixed positive integer and f(z), then f(z) C (k) (λ, α) iff (1 + λjk)(jk + 1 α) a jk+1 + (14) [1 + λ(j 1)] ja j 1 α., j lk+1 heorem 2.5. [6] Let 0 α < 1, 0 λ < 1, k 2 be a fixed positive integer and f(z), then f(z) QC (k) (λ, α) if and only if (jk + 1)(1 + λjk)(jk + 1 α) a jk+1 + (15) [1 + λ(j 1)] j 2 a j 1 α., j lk+1

6 34 On α-uniformly close-to-convex and quasi-convex functions 3 Main results We firstly apply the operator I c+δ (see (4)) on a α-uniformly starlike function of order β with negative coefficients and we prove that the resulting function conserves in the same class of α-uniformly starlike functions of order β with negative coefficients. heorem 3.1. Let F (z) = z a j z j, a j 0, j 2, F (z) US (k) (α, β), α 0, 0 β < 1, k 1 be a fixed positive integer. hen f(z) = I c+δ (F (z)) US (k) (α, β), where I c+δ is the integral operator defined by (4). Proof. From Remark 2.2 we obtain f(z) = I c+δ (F (z)). From Remark 2.1 we have: f(z) = z c + j + δ 1 a jz j, where 0 < c <, j 2, 1 δ <. From F (z) US (k) (α, β), by using heorem 2.2, we have: [(1 + α)(jk + 1) (α + β)] a jk+1 + (16), j lk+1 (1 + α)ja j < 1 β. Using again heorem 2.2 we observe that it is sufficient to prove that: [(1 + α)(jk + 1) (α + β)] c + jk + δ + (17), j lk+1 From hypothesis we have (1 + α)j c + j + δ 1 < 1 β. 0 < c + jk + δ < 1 and 0 < c + j + δ 1 < 1. (18) hus, we see that, by using (16) and (18), the condition (17) holds. means that f(z) US (k) (α, β). his Using a similar method as in heorem 3.1, we apply the operator I c+δ (see (4)) on a α-uniformly convex function of order β with negative coefficients and

7 Irina Dorca 35 we prove that the resulting function conserves in the same class of α-uniformly convex functions of order β with negative coefficients. heorem 3.2. Let F (z) = z a j z j, a j 0, j 2, F (z) UCV (k) (α, β), α 0, 0 β < 1, k 1 be a fixed positive integer. hen f(z) = I c+δ (F (z)) UCV (k) (α, β), where I c+δ is the integral operator defined by (4). Next, we apply the operator I c+δ (see (4)) on a α-uniformly close to convex function of order β with negative coefficients and we prove that the resulting function conserves in the same class of α-uniformly close to convex functions of order β with negative coefficients. heorem 3.3. Let F (z) = z a j z j, a j 0, j 2, F (z) C (k) (α, β), α 0, 0 β < 1, k 1 be a fixed positive integer. hen f(z) = I c+δ (F (z)) C (k) (α, β), where I c+δ is the integral operator defined by (4). Proof. From Remark 2.2 we have f(z) = I c+δ (F (z)). From Remark 2.1 we have: f(z) = z c + j + δ 1 a jz j, where 0 < c <, j 2, 1 δ <. From F (z) C (k) (α, β), by using heorem 2.4, we have: (1 + λjk)(jk + 1 α) a jk+1 + (19), j lk+1 [1 + λ(j 1)]ja j 1 α. Using again heorem 2.4 we notice that it is sufficient to prove that: (1 + λjk)(jk + 1 α) c + jk + δ + (20), j lk+1 [1 + λ(j 1)]j c + j + δ 1 1 α.

8 36 On α-uniformly close-to-convex and quasi-convex functions From hypothesis we have 0 < c + jk + δ < 1 and 0 < c + j + δ 1 < 1. (21) hus, we obtain, by using (19) and (21), that the condition (20) holds. his means that f(z) C (k) (α, β). We end our research by taking into account a similar method as in heorem 3.3, where we apply the operator I c+δ (see (4)) on a quasi-convex function of order β with negative coefficients and we prove that the resulting function conserves in the same class of quasi-convex functions of order β with negative coefficients. heorem 3.4. Let F (z) = z a j z j, a j 0, j 2, F (z) QC (k) (α, β), α 0, 0 β < 1, k 1 be a fixed positive integer. hen f(z) = I c+δ (F (z)) QC (k) (α, β), where I c+δ is the integral operator defined by (4). ACKNOWLEDGEMENS. his work was partially supported by the strategic project POSDRU 107/1.5/S/77265, inside POSDRU Romania co-financed by the European Social Fund-Investing in People. References [1] M. Acu and S. Owa, Note on a class of starlike functions, Proceeding of the International Short Joint Work on Study on Calculus Operators in Univalent Function heory, Kyoto, (2006), [2] G.S. Sălăgean, Geometria Planului Complex, Ed. Promedia Plus, Cluj - Napoca, [3] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 5, (1975),

9 Irina Dorca 37 [4] K. Sakaguchi, In certain univalent mapping, J.M. Soc. Japan, 11, (1959), [5] Wang Zhigang, On Subclasses of Close-to-convex and Quasi-convex Functions With Respect to k-symmetric Points, Advances In Mathematics, 38(1), (2009), [6] Zhi-Gang Wang, Chun-Y Gao, Halit Orfan and Sezgin Akbulut, Some subclasses of close-to-convex and quasi-convex functions with respect to k-symmetric points, General Mathematics, 15(4), (2007),