SUBCLASS OF HARMONIC STARLIKE FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE

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1 LE MATEMATICHE Vol. LXIX (2014) Fasc. II, pp doi: / SUBCLASS OF HARMONIC STARLIKE FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE H. E. DARWISH - A. Y. LASHIN - S. M. SOILEH The purpose of the present paper is to establish some results involving coefficient conditions, distortion bounds, extreme points, convolution, and convex combinations for a new class of harmonic univalent functions in the open unit disc associated with the Salagean operator. We also discuss a class preserving integral operator. Relevant connections of the results presented here with various nown results are briefly indicated. 1. Introduction A continuous complex-valued function f = u + iv is said to be harmonic in a simply connected domain D C if both u and v are real harmonic in D. There is a close inter-relation between analytic functions and harmonic functions. For example, for real harmonic functions u and v there exist analytic functions U and V so that u = R(U) and v = Im(V ). Then f (z) = h(z) + g(z), where h and g are respectively, the analytic functions (U +V )/2 and (U V )/2. In this case, the Jacobian of f = h + g is given by Entrato in redazione: 16 settembre 2013 J f = h (z) 2 g (z) 2. AMS 2010 Subject Classification: 30C45. Keywords: Harmonic, Univalent, Starlie, Convolution.

2 148 H. E. DARWISH - A. Y. LASHIN - S. M. SOILEH The mapping z f (z) is orientation preserving and locally one-to-one in D if and only if J f > 0 in D. The function f = h + g is said to be harmonic univalent in D if the mapping z f (z) is orientation preserving, harmonic and one-to-one in D. We call h the analytic part and g the co-analytic part of f. See Clunie and Sheil-Small [2]. Denote by H the class of functions f = h + g that are harmonic univalent and orientation preserving in the open unit dis U = z : z < 1} for which f (0) = f z (0) 1 = 0. Then for f = h + g H, we may express the analytic functions f and g as h(z) = z + a z, g(z) = b z, b 1 < 1. (1) Note that H reduces to the class S of normalized analytic univalent functions if the co-analytic part of its member is zero. For this class the function f (z) may be expressed as f (z) = z + a z. (2) A function f = h + g with h and g be given by (1) is said to be harmonic starlie of order β,(0 β < 1) for z = r < 1, if } θ (arg f zh (z) zg (reiθ )) = R (z) β. h(z) + g(z) The class of all harmonic starlie functions of order β is denoted by S H (β) and extensively studied by Jahangiri [6]. The case β = 0 and b 1 = 0 were studied by Silverman and Silvia [19] and Silverman [18], for other related wors of the class H, (see also [1], [3], [4], [5], [7], [8], [9], [12], [15], [16]). Definition 1.1. Let D n f = D n h + D n g with h and g be given by (1). Where D n h(z) = z + n a z, D n g(z) = n b z, b 1 < 1. (3) Then f SH,n (α,β) if and only if, for β,(0 β < 1) and α 0, } R αd n+2 h(z) + (1 α)d n+1 h(z) + αd n+2 g(z) + (α 1)D n+1 g(z) D n h(z) + D n g(z) β.

3 S H SUBCLASS OF HARMONIC STARLIKE FUNCTIONS 149 We note that for α = 0 and n = 0, the class SH,n (α,β) reduces to the class (β). Also we note that SH,0(α,β) = SH(α,β) = f (z) H : R αz 2 h (z)+zh (z)+αz 2 g (z)+(2α 1)zg (z) h(z)+g(z) } } β. Further, if the co-analytic part g(z) is zero the class SH,n (α,β) reduces to the class P n (α,β) of function f S which satisfy the condition } az 2 (D n f ) + z(d n f ) R D n β f (z) for some β(0 β < 1),α 0, f (z) z 0 and z U. Observe that the classes P 0 (α,β) and P 0 (α,0) were introduced and studied by many authors such as Obradovic and Joshi [13], Padmanabhan [14], Liu and Owa [11], Xu and Yang [21], Singh and Gupta [20] and Lashin [10]. We also note that for α = 0, the class P 0 (0,β) was studied by Silverman [17]. 2. Main Results We begin with the statement of the following lemma due to Jahangiri [6]. Lemma 2.1. Let f = h + g with h and g of the form (1). Furthermore, let β a + + β b 1 where 0 β < 1. Then f is harmonic, orientation preserving, univalent in U, and f S H (β). Theorem 2.2. Let f = h + g with h and g of the form (1). If n α( 1) + β a + n α( + 1) + + β b 1 (4) for some β,(0 β < 1) and α 0, then f is harmonic, orientation preserving, univalent in U and f S H,n (α,β). Proof. Since β n α( 1) + β and + β n α( + 1) + + β,( 1),

4 150 H. E. DARWISH - A. Y. LASHIN - S. M. SOILEH it follows from Lemma 2.1, that f SH (β) and hence f is harmonic, orientation preserving and univalent in U. Now, we only need to show that if (4) holds then } αd n+2 h(z) + (1 α)d n+1 h(z) + αd R n+2 g(z) + (α 1)D n+1 g(z) D n h(z) + D n g(z) = R A(z) B(z) β. Using the fact that R(w) β if and only if + w 1 + β w, it suffices to show that where and A(z) + ()B(z) A(z) (1 + β)b(z) 0, (5) A(z) = αd n+2 h(z) + (1 α)d n+1 h(z) + αd n+2 g(z) + (α 1)D n+1 g(z) B(z) = D n h(z) + D n g(z) Substituting for A(z) and B(z) in (5), we obtain A(z) + ()B(z) A(z) (1 + β)b(z) = (2 β)z + n [(α + 1)( 1) + 2 β]a z + n [α( + 1) ( 1 + β)]b z βz + n [(α + 1)( 1) β]a z + n [α( + 1) ( β)]b z (2 β) z n [(α + 1)( 1) + 2 β] a z n α( + 1) ( 1 + β) b z β z n [(α + 1)( 1) β] a z n α( + 1) ( β) b z

5 SUBCLASS OF HARMONIC STARLIKE FUNCTIONS 151 2() z 2 2() z 1 by the given condition (4). The Harmonic function f (z) = z + n [α( 1) + β] a z 2 n [α( 1)+ β] 1 β a n [α( 1) + β] x z + n [α( + 1) + ( + β)] b z n [α(+1)++β] 1 β b } 0, n [α( + 1) + + β] y z (6) where x + y = 1, show that the coefficient bound given in Theorem 2.2 is sharp. The functions of the form (6) are in SH.n (α,β) since n [α( 1) + β] a + n [α( + 1) + + β] b = x + y = 1. We note that: (i) when n = 0 and α = 0 in Theorem 2.2, we obtain the result obtained by Jahangiri [[6], Theorem 1]; (ii) when n = 0 and g(z) = 0 in Theorem 2.2, we obtain the result obtained by Lashin [[10], Theorem 2.1]. We denote by S H,n(α,β) to the class of functions f SH,n (α,β) whose coefficients satisfy the condition (4). Theorem 2.3. Let 0 α 1 < α 2 and 0 β < 1. Then S H,n(α 2,β) S H,n(α 1,β). Proof. It follows from (4) that n [α 1( 1) + β] < a + n [α 2( 1) + β] n [α 1( + 1) + + β] b a + for f S H,n(α 2,β). Hence f S H,n(α 1,β). n [α 2( + 1) + + β] b 1

6 152 H. E. DARWISH - A. Y. LASHIN - S. M. SOILEH Theorem 2.4. Let α > 0, 0 β < 1. Then S H,n+1(α,β) S H,n(α,β) The proof follows immediately from Theorem 2.2. In the following Corollary we show that the functions in S H,n(α,β) are starlie harmonic in U. Corollary 2.5. For 0 α and 0 β < 1, S H,n(α,β) S H(β). The proof is now immediate from Theorems 2.3 and Distortion Bounds and Extreme Points Now we obtain the distortion bounds for functions in the class S H,n(α,β). Theorem 3.1. Let f = h + g with h and g of the form (1) and f S H,n(α,β). Then for z = r < 1, we have f (z) (1 + b 1 )r + 1 [ 2 n 2α + 2 β 2α β ] 2α + 2 β b 1 r 2 (7) f (z) (1 b 1 )r 1 [ 2 n 2α + 2 β 2α β ] 2α + 2 β b 1 r 2 (8) where The result is sharp. b 1 2α β. Proof. We shall prove the first inequality. Let f S H,n(α,β). Then we have and so f (z) (1 + b 1 )r + = (1 + b 1 )r + 2 n [2α + 2 β] + 2 n [2α + 2 β] f (z) (1 + b 1 )r ( a + b )r (1 + b 1 )r + r 2 ( a + b ) n [ α( 1) + β (1 + b 1 )r + 2 n [2α + 2 β] 2 n [2α + 2 β] ( a + b )r 2 α( + 1) + + β a + [ 1 2α β ] b r 2 ] b 1 ] = (1 + b 1 )r + 1 [ 2 n 2α + 2 β 2α β 2α + 2 β b 1 r 2. r 2

7 SUBCLASS OF HARMONIC STARLIKE FUNCTIONS 153 The proof of the inequality (8) is similar, thus we omit it. The upper bound given for f S H,n(α,β)) is sharp and the equality occurs for the function f (z) = z + b 1 z + 1 [ 2 n 2α + 2 β 2α β ] 2α + 2 β b 1 z 2 (z = r), b 1 2α β. This completes the proof of Theorem 3.1. Now we determine the extreme points of the closed convex hull of the class S H,n(α,β) denoted by clcohs H,n(α,β). Theorem 3.2. Let f = h + g, where h and g are given by (1). clcohs H,n(α,β) if and only if Then f where f (z) = (X h +Y g ) (9) h 1 (z) = z h (z) = z + n [α( 1) + β] z ( = 2,3,...), g (z) = z + n [α( + 1) + + β] z ( = 1,2,3,...), (X +Y ) = 1, X 0 and Y 0. In particular, the extreme points of the class S H,n(α,β) are h } and g } respectively. Proof. For a function f of the form (9), we have f (z) = = = z + (X h +Y g ) (X +Y )z + n [α( 1) + β] X z + n [α( 1) + β] X z + n [α( + 1) + + β] Y z n [α( + 1) + + β] Y z.

8 154 H. E. DARWISH - A. Y. LASHIN - S. M. SOILEH But = n [ [α( 1) + β] X + + ] n [α( 1) + β] X [ n [α( + 1) + + β] Y = 1 X 1 1. n [α( + 1) + + β] X ] and Thus f clcohs H,n(α,β). Conversely, suppose that f clcohs H,n(α,β). Set X = n [α( 1) + β] a ( = 2,3,...), (10) Y = n [α( + 1) + + β] b ( = 1,2,3,...). (11) Then by the inequality (4), we have 0 X 1( = 2,3,...) and 0 Y 1( = 1,2,...). Define X 1 = 1 X Y and note that X 1 0. Thus we obtain f (z) = X h +Y g. This completes the proof of Theorem Convolution and Convex Combinations For two harmonic functions and we define their convolution f (z) = z + F(z) = z + ( f F)(z) = z + a z + A z + a A z + b z B z b B z, using this definition, we show that the class S H,n(α,β) is closed under convolution. Theorem 4.1. For some 0 β < 1 and α 0, let f,f S H,n(α,β). Then f F S H,n(α,β).

9 SUBCLASS OF HARMONIC STARLIKE FUNCTIONS 155 Proof. We note that A 1 and B 1. Now for the convolution ( f F) we have n [α( 1) + β] A a + n [α( 1) + β] a + n [α( + 1) + + β] B b n [α( + 1) + + β] b 1. Therefore f F S H,n(α,β). We show that the class S H,n(α,β) is closed under convex combination of its members. Theorem 4.2. The class S H,n(α,β) is closed under convex combination. Proof. For (i = 1,2,3...) let f i S H,n(α,β) where f i (z) is given by Then by (4), we have f i (z) = z + n [α( 1) + β] a i + a i z + b i z. n [α( + 1) + + β] b i 1. For i=1 t i = 1, 0 t i 1, the convex combination of f i may be written as i=1 Then by (4), we have t i f i = z + n [α( 1) + β] i=1t i ( ( i=1 t i a i )z + ( i=1 t i b i )z. n [α( + 1) + + β] t i a i + i=1 i b i i=1t ) n [α( 1) + β] n [α( + 1) + + β] a i + b i Therefore i=1 t i f i S H,n(α,β). i=1 t i = 1.

10 156 H. E. DARWISH - A. Y. LASHIN - S. M. SOILEH 5. A Family of Class Preserving Integral Operator Let f (z) = h(z) + g(z) H be given by(1) then F(z) defined by relation F(z) = c + 1 z z c t c 1 h(t)dt + c + 1 z 0 z c t c 1 g(t)dt, (c > 1) (12) 0 Theorem 5.1. Let f = h + g, where h and g are given by (1) be in the class S H,n(α,β), then F(z) defined by (12) also belong to the class S H,n(α,β). Proof. From the representation of F, it follows that Now F(z) = z + n [α( 1) + β] + c + 1 c + a z + ( ) c + 1 c + a n [α( + 1) + + β] n [α( 1) + β] a + by (4). Thus F(z) S H,n(α,β). c + 1 c + b z. ( ) c + 1 c + b n [α( + 1) + + β] b 1 Acnowledgements The authors would lie to than the referee for his/her careful reading and maing some valuable comments which have essentially improved the presentation of this paper. REFERENCES [1] O. P. Ahuja - J. M. Jahangiri, A subclass of harmonic univalent functions, J. of Nat. Geom. 20 (2001), [2] J. Clunie - T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fen. Series AI Math. 9 (3) (1984), [3] R. M. El-Ashwah - M. K. Aouf - A. M. Hassan - A. H. Hassan, A Unified representation of some starlie and convex harmonic functions with negative coefficients, Opuscula Math. 33 (2) (2013),

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12 158 H. E. DARWISH - A. Y. LASHIN - S. M. SOILEH H. E. DARWISH Department of Mathematics Faculty of Science Mansoura University, Mansoura 35516, Egypt Darwish333@yahoo.com A. Y. LASHIN Department of Mathematics Faculty of Science Mansoura University, Mansoura 35516, Egypt aylashin@mans.edu.eg S. M. SOILEH Department of Mathematics Faculty of Science Mansoura University, Mansoura 35516, Egypt s soileh@yahoo.com