SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH THE DERIVATIVE OPERATOR

Transcription

1 Hacettepe Journal of Mathematics and Statistics Volume 41(1) (2012), SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH THE DERIVATIVE OPERATOR A.L. Pathak, S.B. Joshi, Preeti Dwivedi and R. Agarwal Received 31:07:2010 : Accepted 24:08:2011 Abstract In the present paper, we introduce a new subclass of harmonic functions in the unit disc U by using the Derivative operator. Also, we obtain coefficient conditions, convolution conditions, convex combinations, extreme points and some other properties. Keywords: Harmonic, Univalent functions, Derivative operator AMS Classification: 30C45, 31A Introduction A continuous function f = uiv is a complex-valued harmonic function in a complex domain C if both u and v are real harmonic in C. In any simply connected domain D C, we can write f = hg, where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f. A necessary and sufficient condition for f to be locally univalent and sense-preserving in D is that h (z) > g (z) in D, see 4]. In 1984, Clunie and Sheil-Small 4] investigated the class S H and studied some sufficient bounds. Since then, there have been several papers published related to S H and its subclasses. In fact by introducing new subclasses Sheil-Small 13], Silverman 14], Silverman and Silvia 15], Jahangiri 6] and Ahuja 1] presented a systematic and unified study of harmonic univalent functions. Furthermore we refer to Duren 5], Ponnusamy 9] and references therein for basic results on the subject. Department of Mathematics, Brahmanand College, The Mall, Kanpur , (U.P.), India. (A. L. Pathak) alpathak@rediffmail.com alpathak.bnd@gmail.com (R. Agarwal) ritesh.840@ rediffmail.com Department of Mathematics, Walchand College of Engineering, Sangli , (Maharashtra), India. joshisb@hotmail.com Corresponding Author. Department of Physics, V.S.S.D. College, Kanpur , (U.P.), India. priti vdwivedi@ yahoo.com

2 48 A.L. Pathak, S.B. Joshi, P. Dwivedi, R. Agarwal Denote by S H, the class of functions f = h g that are harmonic, univalent and sense-preserving in the unit disk U = {z : z < 1} with normalization f(0) = h(0) = f z(0) 1 = 0. Then for f = hg S H, we may express the analytic functions h and g as (1.1) h(z) = z a k z k, g(z) = b k z k, b 1 < 1. Observe that S H reduces to S, the class of normalized univalent functions, if the coanalytic part of f is zero. Also, denote by S H the subclass of S H consisting of functions f that map U onto a starlike domain. For f = hg given by (1.1), Al-Shaqsi and Darus 3] introduced the operator D n λ as: (1.2) Dλf(z) n = Dλh(z)( 1) n n Dλ n g(z), n,λ N0 = N {0}, z U, where Dλh(z) n = z k n C(λ,k)a k z k, Dλg(z) n = k n C(λ,k)b k z k and C(λ,k) = ( kλ 1 ) λ. Recently Rosy et al. 10] defined the subclass G H(γ) S H consisting of harmonic univalent functions f(z) satisfying the condition { } Re (1e iα ) zf (z) f(z) eiα γ, 0 γ < 1, α R. They proved that if f = hg is given by (1.1) and if (2n 1 γ) (1.3) a (2n1γ) ] n b n 2, 0 γ < 1, (1 γ) (1 γ) n=1 then f is in G H(γ). This condition is proved to be also necessary by Rosy et al. if h and g are of the form (1.4) h(z) = z a n z n, g(z) = b n z n. n=2 n=1 Motivated by this aforementioned work, now we introduce the class G H(n,λ,α,ρ) as the subclass of functions of the form (1.1) that satisfy the following condition } (1.5) Re {(1ρe ir ) Dn1 λ f(z) Dλ nf(z) ρe ir > α, 0 α < 1, r R, ρ 0, where D n λf(z) is defined by (1.2). Let G H(n,λ,α,ρ) denote that the subclasses of G H(n,λ,α,ρ) which consists of harmonic functions f n = hg n such that h and g n are of the form (1.6) h(z) = z a k z k, g n(z) = ( 1) n b k z k. It is clear that the class G H(n,λ,α,ρ) includes a variety of well-known subclasses of S H, such as, (i) G H(0,0,α,0) S H(α), Jahangiri 6], (ii) G H(0,1,α,0) HK(α), Jahangiri 6], (iii) G H(n,0,α,0) MH(n,0,α), Jahangiri et al. 7],

3 Harmonic Univalent Functions 49 (iv) G H(0,λ,α,0) M H (0,λ,α), Murugusundaramoorthy and Vijya 8], (v) G H(n,λ,α,0) M H (n,λ,α), Al-Shaqsi and Darus 2], (vi) G H(0,1,γ,1) G H(γ),Rosy et al. 10]. In this paper, we will give sufficient condition for functions f = h g, where h and g are given by (1.1), to be in the class G H(n,λ,α,ρ) and it is shown that this coefficient condition is also necessary for functions in the class G H(n,λ,α,ρ). Also, we obtain distortion theorems and characterize the extreme points and convolution conditions for functions in G H(n,λ,α,ρ). Closure theorems and an application of neighborhoods are also obtained. 2. Coefficient bound We begin with a sufficient coefficient condition for functions in G H(n,λ,α,ρ) Theorem. Let f = hg be given by (2.1). If (2.1) {k(1ρ) (αρ)} a k {k(1ρ)(αρ)} b k ] k n C(λ,k) 2(), where a 1 = 1, n,λ N 0,C(λ,k) = ( ) kλ 1 λ, ρ 0 and 0 α < 1, then f is sensepreserving, harmonic univalent in U and f G H(n,λ,α,ρ). Proof. If z 1 z 2, then (2.2) f(z1) f(z2) h(z 1) h(z 2) 1 g(z 1) g(z 2) h(z 1) h(z 2) b k (z1 k z k 2) = 1 (z 1 z 2) a k (z1 k z2) k > 1 1 0, 1 1 k b k k a k k(1ρ)(αρ)]k n C(λ,k) b k k(1ρ) (αρ)]k n C(λ,k) a k

4 50 A.L. Pathak, S.B. Joshi, P. Dwivedi, R. Agarwal which proves univalence. Note that f is sense-preserving in U. This is because h (z) 1 k a k z k 1 (2.3) > 1 > {k(1ρ) (αρ)}k n C(λ,k) a k {k(1ρ)(αρ)}k n C(λ,k) b k {k(1ρ)(αρ)}k n C(λ,k) b k z k 1 k b k z k 1 g (z). Using the fact that Rew > α if and only if w 1α w it suffices to show that (2.4) ()(1ρe ir ) Dn1 λ f(z) Dλ nf(z) ρe ir (1α) (1ρe ir ) Dn1 λ f(z) Dλ nf(z) ρe ir 0. Substituting the value of D n λf(z) in (2.4) yields, by (2.1), ( ρe ir )Dλf(z)(1ρe n ir )D n1 λ f(z) (1αρe ir )Dλf(z)(1ρe n ir )D n1 λ f(z) = (2 α)z {k(1ρe ir )( ρe ir )}k n C(λ,k) a k z k ( 1) n {k(1ρe ir ) ( ρe ir )}k n C(λ,k)b k z k αz {k(1ρe ir ) (1αρe ir )}k n C(λ,k)a k z k ( 1) n {k(1ρe ir )(1αρe ir )}k n C(λ,k)b k z k {k(1ρ) (αρ)}k n C(λ,k) a k z k 2() z 1 ] {k(1ρ)(αρ)}k n C(λ,k) b k z k (2.5) 2() 1 {k(1ρ) (αρ)}k n C(λ,k) a k ]. {k(1ρ)(αρ)}k n C(λ,k) b k This last expressions is non-negative by (2.1), and so the proof is complete.

5 Harmonic Univalent Functions 51 The harmonic function () f(z) = z {k(1ρ) (αρ)}k n C(λ,k) x kz k (2.6) () {k(1ρ)(αρ)}k n C(λ,k) y kz k where n,λ N 0, o ρ 1 and x k y k = 1 shows that the coefficient bound given by (2.1) is sharp. The functions of the form (2.6) are in G H(n,λ,α,ρ) because k(1ρ) (αρ) a k k(1ρ)(αρ) ] b k k n C(λ,k) (2.7) = 1 x k y k = 2. In the following theorem, it is shown that the condition(2.1) is also necessary for functions f n = hg n, where h and, g n are of the form (1.6) Theorem. Let f n = hg n be given by (1.6). Then f n G H(n,λ,α,ρ) if and only if (2.8) {k(1ρ) (αρ)} a k {k(1ρ)(αρ)} b k ]k n C(λ,k) 2() where a 1 = 1, n,λ N 0, C(λ,k) = ( ) kλ 1 λ,ρ 0,0 α < 1. Proof. Since G H(n,λ,α,ρ) G H(n,λ,α,ρ) we only need to prove the only if part of Theorem 2.2. To this end, for functions f n of the form (1.6), we notice that the condition (1.5) is equivalent to } Re {(1ρe ir ) Dn1 λ f(z) Dλ nf(z) (ρe ir α) 0 = Re {(1ρeir )D n1 λ f(z) (ρe ir α)dλf(z)} n Dλ nf(z) 0 = ( (1ρe ir ) z k n1 C(λ,k) a k z k ( 1) 2n1 k n1 b k C(λ,k)z k) Re z k n C(λ,k) a k z k ( 1) 2n k n C(λ,k) b k z k (ρe ir α)(z k n C(λ,k) a k z k ( 1) 2n k n b k C(λ,k)z k ) 0 z k n C(λ,k) a k z k ( 1) 2n k n C(λ,k) b k z k

6 52 A.L. Pathak, S.B. Joshi, P. Dwivedi, R. Agarwal = ()z k n k(1ρe ir ) (ρe ir α)]c(λ,k) a k z k Re z k n C(λ,k) a k z k ( 1) 2n k n C(λ,k) b k z k ( 1) 2n1 k n k(1ρe ir )(ρe ir α)]c(λ,k) b k z k 0 z k n C(λ,k) a k z k ( 1) 2n k n C(λ,k) b k z k = (2.9) () k n k(1ρe ir ) (ρe ir α)]c(λ,k) a k z k 1 Re 1 k n C(λ,k) a k z k 1 z z ( 1)2n k n C(λ,k) b k z k 1 z z ( 1)2n k n k(1ρe ir )(ρe ir α)]c(λ,k) b k z k 1 1 k n C(λ,k) a k z k 1 z 0 z ( 1)2n k n C(λ,k) b k z k 1 The above condition (2.9) must hold for all values of z on the positive real axes, where, 0 z = γ < 1, we must have () k n (k α)c(λ,k) a k γ k 1 Re 1 k n C(λ,k) a k γ k 1 ( 1) 2n k n C(λ,k) b k γ k 1 ( 1) 2n k n (k α)c(λ,k) b k γ k 1 ρe ir k n (k 1)C(λ,k) a k γ k 1 1 k n C(λ,k) a k γ k 1 ( 1) 2n k n C(λ,k) b k γ k 1 ( 1) 2n ρe ir k n (k 1)C(λ,k) b k γ k k n C(λ,k) a k γ k 1 ( 1) 2n k n C(λ,k) b k γ k 1

7 Harmonic Univalent Functions 53 Since Re( e ir ) e ir = 1, the above inequality reduce to (2.10) () k n {(k(1ρ) (ρα))}c(λ,k) a k γ k 1 1 k n C(λ,k) a k γ k 1 k n C(λ,k) b k γ k 1 k n {(k(1ρ)(ρα))}c(λ,k) b k γ k k n C(λ,k) a k γ k 1 k n C(λ,k) b k γ k 1 If the condition (2.8) does not hold, then the numerator in (2.10) is negative for γ sufficiently close to 1. Hence there exists a z 0 = γ 0 in (0,1) for which the quotient in (2.10) is negative. This contradicts the condition for f n G H(n,λ,α,ρ) and so the proof is complete. 3. Distortion bounds In this section, we will obtain distortion bounds for functions in G H(n,λ,α,ρ) Theorem. Let f n G H(n,λ,α,ρ). Then for z = γ < 1, we have () f n(z) (1 b 1 )γ 1 12ρα ] 2 n 2(1ρ) (ρα)](λ1) b1 γ 2. () f n(z) (1 b 1 )γ 1 12ρα ] 2 n 2(1ρ) (ρα)](λ1) b1 γ 2. Proof. We only prove the left-hand inequality. The proof for the right-hand inequality is similar and is thus omitted. Let f n G H(n,λ,α,ρ). Taking the absolute value of f n, we obtain f n(z) = z a k z k ( 1) n b k z k (1 b 1 )γ (1 b 1 )γ ( a k b k )γ k ( a k b k )γ 2 (1 b 1 )γ (2(1ρ) (ρα))2 n (λ1) ( (2(1ρ) (ρα))2 n (λ1) a k (2(1ρ) (ρα))2n (λ1) b k ) γ 2

8 54 A.L. Pathak, S.B. Joshi, P. Dwivedi, R. Agarwal The functions (1 b 1 )γ (2(1ρ) (ρα))2 n (λ1) (k(1ρ) (ρα))k n C(λ,k) a k (1 b 1 )γ (1 b 1 )γ f(z) = z b 1 z (k(1ρ)(ρα))kn C(λ,k) b k (2(1ρ) (ρα))2 n (λ1) (2(1ρ) (ρα))2 n (λ1) 1 2 n (λ1) f(z) = (1 b 1 )z 1 2 n (λ1) ] γ 2 1 ((1ρ)(ρα)) ] b ρα ] b1 γ 2. ] 2(1ρ) (ρα) 12ρα 2(1ρ) (ρα) b1 z 2, ] 2(1ρ) (ρα) 12ρα 2(1ρ) (ρα) b1 z 2 for b 1 show that the bounds given in Theorem 3.1 are sharp. 12ρα The following covering result follows from the left-hand inequality in Theorem Corollary. If the function f n = h g n, where h and g given by (1.4) are in G H(n,λ,α,ρ), then { w : w < (2n (λ1)(ρ2) 1 (2 n (λ1) 1)α) 2 n (λ1)(2(1ρ) (ρα)) (3.1) } 2n (λ1)(ρ2) (2ρ1) (2 n (λ1)1)α b 1 f n(u) 2 n (λ1)(2(1ρ) (ρα)) 4. Convolution, convex combinations and extreme points In this section, we show the class G H(n,λ,α,ρ) is invariant under convolution and convex combination. and For harmonic functions f n(z) = z a k z k ( 1) n b k z k F n(z) = z A k z k ( 1) n B k z k, the convolution of f n and F n is given by (4.1) (f n F n)(z) = f n(z) F n(z) = z a k A k z k ( 1) n b k B k z k. γ 2

9 Harmonic Univalent Functions Theorem. For 0 β α < 1, let f n G H(n,λ,α,ρ) and F n G H(n,λ,β,ρ). Then f n F n G H(n,λ,α,ρ) G H(n,λ,β,ρ). Proof. We wish to show that the coefficient of f n F n satisfies the required condition given in Theorem 2.2. For F n G H(n,λ,β,ρ), we note that A k 1 and B k 1. Now, for the convolution function f n F n, we obtain {k(1ρ) (β ρ)}k n C(λ,k) a k A k 1 β {k(1ρ)(β ρ)}k n C(λ,k) b k B k 1 β {k(1ρ) (β ρ)}k n C(λ,k) a k 1 β {k(1ρ)(β ρ)}k n C(λ,k) b k 1 β 1. {k(1ρ) (αρ)}k n C(λ,k) a k {k(1ρ)(αρ)}k n C(λ,k) b k Since 0 β α < 1 and f n G H(n,λ,α,ρ), then f n F n G H(n,λ,α,ρ) G H(n,λ,β,ρ). We now examine convex combinations of G H(n,λ,α,ρ). Let the functions f nj (z) be defined, for j = 1,2,...,m, by (4.2) f nj (z) = z a k,j z k ( 1) n b k,j z k Theorem. Let the functions f nj (z) defined by (4.2) be in the class G H(n,λ,α,ρ) for every j = 1,2,...,m. Then the functions t j(z) defined by (4.3) t j(z) = m c jf nj (z), 0 c j 1, are also in the class G H(n,λ,α,ρ), where m cj = 1. Proof. According to the definition of t j, we can write ( m ) ( m ) t j(z) = z c j a k,j z k ( 1) n c j b n,j z k.

10 56 A.L. Pathak, S.B. Joshi, P. Dwivedi, R. Agarwal Further, since f nj (z) are in G H(n,λ,α,ρ) for every j = 1,2,...,m, then { ( m ) (k(1ρ) (αρ)) c j a k,j = ( m )] } (k(1ρ)(αρ)) c j b k,j k n C(λ,k) m ( c j (k(1ρ) (αρ)) a n,j ) (k(1ρ)(αρ)) b n,j ]k n C(λ,k) m c j2() 2(). Hence Theorem 4.2 follows Corollary. The class G H(n,λ,α,ρ) is closed under convex linear combinations. Proof. Letthefunctionsf nj (z)(j = 1,2...,m)definedby(4.2)beintheclassG H(n,λ,α,ρ). Then the function Ψ(z) defined by (4.4) Ψ(z) = µf nj (z)(1 µ)f nj (z), 0 µ 1 is intheclass G H(n,λ,α,ρ). Also, bytakingm = 2, t 1 = µ andt 2 = 1 µ in Theorem 4.1. Next we determine the extreme points of closed convex hulls of G H(n,λ,α,ρ), denoted by clcog H(n,λ,α,ρ) Theorem. Let f n be given by (1.6). Then f n G H(n,λ,α,ρ) if and only if f n(z) = (X k h k (z)y k g nk (z)), where and ( h 1(z) = z, h k (z) = z ( g nk (z) = z ( 1) n (k(1ρ) (αρ))k n C(λ,k) ) z k, k = 2,3..., ) z k, k = 1,2,3... (k(1ρ)(αρ))k n C(λ,k) (X k Y k ) = 1, X k 0, Y k 0. In particular, the extreme points of G H(n,λ,α,ρ) are {h k } and {g nk }. Proof. For the function f n of the form (4.7), we have f n(z) = (X k h k (z)y k g nk (z)) = (X k Y k )z (k(1ρ) (αρ))k n C(λ,k) X kz k ( 1) n (k(1ρ)(αρ))k n C(λ,k) Y kz k.

11 Harmonic Univalent Functions 57 Then (4.5) (k(1ρ) (αρ))k n C(λ,k) a k and so f n clcog H(n,λ,α,ρ). (k(1ρ)(αρ))k n C(λ,k) b k = X k Y k = 1 X 1 1, Conversely, suppose that f n clcog H(n,λ,α,ρ). Setting X k = (k(1ρ) (αρ))kn C(λ,k) a k, 0 X k 1 k = 2,3,..., (4.6) Y k = (k(1ρ)(αρ))kn C(λ,k) b k, 0 Y k 1 k = 1,2,3,..., and X 1 = 1 X k Y k then f n can be written as (4.7) f n(z) = z = z = z a k z k ( 1) n b k z k ()X k (k(1ρ) (αρ))k n C(λ,k) zk ( 1) n ()Y k (k(1ρ)(αρ))k n C(λ,k) zk (h k (z) z)x k (g nk (z) z)y k ( ) = h k (z)x k g nk (z)y k z 1 X k Y k = (h k (z)x k g nk (z)y k ), as required. Using Corollary 4.3 we have clcog H(n,λ,α,ρ) = G H(n,λ,α,ρ). Then the statement of Theorem 4.4 is true for f G H(n,λ,α,ρ). Acknowledgement The authors are thankful to Prof. K. K. Dixit, Department of Mathematics, Gwalior Engineering College, Gwalior (M.P.), India for giving us his fruitful suggestions and comments. References 1] Ahuja, O. P. Planar harmonic univalent and related mappings, J. Inequal. Pure Appl. Math. 6(4), Art. 122, 1 18, ] Al-Shaqsi, K. and Darus, M. On harmonic functions defined by derivative operator, Journal of Inequalities and Applications, Art. ID , 1 10, 2008.

12 58 A.L. Pathak, S.B. Joshi, P. Dwivedi, R. Agarwal 3] Al-Shaqsi, K. and Darus, M. An operator defined by convolution involving the polylogarithms functions, Journal of Math. and Stat. 4(1), 46 50, ] Clunie, J. and Sheil-Small, T. Harmonic univalent functions, Ann. Acad. Sci. Fen. Series AI Math. 9(3), 3 25, ] Duren, P. Harmonic Mappings in the Plane (Cambridge Univ. Press, Cambridge, UK, 2004). 6] Jahangiri, J. M. Harmonic functions starlike in the unit disc, J. Math. Anal. Appl. 235, , ] Jahangiri, J. M., Murugusundaramoorthy, G. and Vijaya, K. Salagean-type harmonic univalent functions, Southwest J. Pure Appl. Math. 2, 77 82, ] Murugusundaramoorthy, G. and Vijaya, K. On certain classes of harmonic univalent functions involving Ruscheweyh derivatives, Bull. Cal. Math. Soc. 96(2), , ] Ponnusamy, S. and Rasila, A. Planar harmonic mappings, RMS Mathematics Newsletter 17(2) (2007), 40 57, ] Rosy, T., Stephen, B. A., Subramanian, K. G. and Jahangiri, J. M. Goodman-Ronning-type harmonic univalent functions, Kyungpook Math. J. 41(1), 45 54, ] Ruscheweyh, S. Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81, , ] Ruscheweyh, S. Convolutions in Geometric Function Theory (Les Presses de I Universite de Montreal, 1982). 13] Sheil-Small, T. Constants for planar harmonic mappings, J. London Math. Soc. 2(42), , ] Silverman, H. Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl. 220, , ] Silverman, H. and Silvia, E. M. Subclasses of harmonic univalent functions, New Zealand J. Math. 28, , ] Srivastava, H. M. and Owa, S. Current Topics in Analytic Function Theory (World Scientific Publishing Company, Singapore, 1992).