Thông tin tài liệu
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 263413, 10 pages doi:10.1155/2008/263413 Research Article On Harmonic Functions Defined by Derivative Operator K. Al-Shaqsi and M. Darus School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D. Ehsan, Malaysia Correspondence should be addressed to M. Darus, maslina@pkrisc.cc.ukm.my Received 16 September 2007; Revised 20 November 2007; Accepted 26 November 2007 Recommended by Vijay Gupta Let S H denote the class of functions f h –– g that are harmonic univalent and sense-preserv- ing in the unit disk U {z : |z| < 1}, where hzz ∞ k2 a k z k ,gz ∞ k1 b k z k |b 1 | < 1. In this paper, we introduce the class M H n, λ,α of functions f h –– g which are harmonic in U. Asufficient coefficient of this class is determined. It is shown that this coefficient bound is also nec- essary for the class M –– H n, λ,α if f n zh –– g n ∈ M H n, λ,α, where hzz − ∞ k2 |a k |z k ,g n z −1 n ∞ k1 |b k |z k and n ∈ N 0 .Coefficient conditions, such as distortion bounds, convolution con- ditions, convex combination, extreme points, and neighborhood for the class M –– H n, λ, α,areob- tained. Copyright q 2008 K. Al-Shaqsi and M. Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction A continuous function f u iv 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,wecanwrite f h g,whereh and g are analytic in D.Wecallh the analytic part and g the coanalytic 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 2. 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} for which f0h0f z 0 − 1 0. Then for f h g ∈S H , we may express the analytic functions h and g as hzz ∞ k2 a k z k ,gz ∞ k1 b k z k , b 1 < 1. 1.1 2 Journal of Inequalities and Applications Observe that S H reduces to S, the class of normalized univalent analytic functions, if the coan- alytic part of f is zero. Also, denote by S ∗ H the subclasses of S H consisting of functions f that map U onto starlike domain. For f h g given by 1.1, we define the derivative operator introduced by authors see 1 of f as D n λ fzD n λ hz−1 n D n λ gz ,n,λ∈ N 0 N ∪{0},z∈ U, 1.2 where D n λ hzz ∞ k2 k n Cλ, ka k z k , D n λ gz ∞ k1 k n Cλ, kb k z k , and Cλ, k kλ−1 λ . We let M H n, λ, α denote the family of harmonic functions f of the form 1.1 such that Re D n1 λ fz D n λ fz >α, 0 ≤ α<1, 1.3 where D n λ f is defined by 1.2. If the coanalytic part of f h g is identically zero, then the class M H n, λ, α turns out to be the class R n λ α introduced by Al-Shaqsi and Darus 1 for the analytic case. Let M H n, λ, α denote that the subclass of M H n, λ, α consists of harmonic functions f n h g n such that h and g n are of the form hzz − ∞ k2 a k z k ,g n z−1 n ∞ k1 b k z k . 1.4 It is clear that the class M H n, λ, α includes a variety of well-known subclasses of S H .For example, M H 0, 0,α ≡ S ∗ H α is the class of sense-preserving, harmonic, univalent functions f which are starlike of order α in U,thatis,∂/∂θ argfre iθ >α,andM H 1, 0,α ≡ M H 0, 1,α ≡HKα is the class of sense-preserving, harmonic, univalent functions f which are convex of order α in U,thatis,∂/∂θ arg∂/∂θfre iθ >α. Note that the classes S ∗ H and HKα were introduced and studied by Jahangiri 3. Also we notice that the class M H n, 0,α is the class of Salagean-type harmonic univalent functions introduced by Jahangiri et al. 4;andM H 0,λ,α is the class of Ruscheweyh-type harmonic univalent functions stud- ied by Murugusundaramoorthy and Vijaya 5. In 1984, Clunie and Sheil-Small 2 investigated the class S H as well as its geometric sub- classes and obtained some coefficient bounds. Since then, there has been several related papers on S H and its subclasses such that Silverman 6, Silverman and Silvia 7, and Jahangiri 3, 8 studied the harmonic univalent functions. Jahangiri and Silverman 9 prove the following theorem. Theorem 1.1. Let f h g given by 1.1.If ∞ k2 k a k b k ≤ 1 − b 1 , 1.5 then f is sense-preserving, harmonic, and univalent in U and f ∈ S ∗ H consists of functions in S H which are starlike in U. The condition 1.5 is also necessary if f ∈TH ≡ M H 0, 0, 0. In this paper, we will give sufficient condition for functions f h g,whereh and g are given by 1.1 to be in the class M H n, λ, α; and it is shown that this coe fficient condition is K. Al-Shaqsi and M. Darus 3 also necessary for functions in the class M H n, λ, α. Also, we obtain distortion theorems and characterize the extreme points for functions in M H n, λ, α. Closure theorems and application of neighborhood are also obtained. 2. Coefficient bounds We begin with a sufficient coefficient condition for functions in M H n, λ, α. Theorem 2.1. Let f h g be given by 1.1.If ∞ k1 k − α a k k α b k k n Cλ, k ≤ 21 − α, 2.1 where a 1 1,n,λ∈ N 0 ,Cλ, k kλ−1 λ , and 0 ≤ α<1,thenf is sense-preserving, harmonic, univalent in U,andf ∈ M H n, λ, α. Proof. If z 1 / z 2 ,then f z 1 − f z 2 h z 1 − h z 2 ≥ 1 − g z 1 − g z 2 h z 1 − h z 2 1 − ∞ k1 b k z k 1 − z k 2 z 1 − z 2 ∞ k2 a k z k 1 − z k 2 >1− ∞ k1 k b k 1 − ∞ k2 k a k ≥1− ∞ k1 k αk n Cλ, k/1 − α b k 1− ∞ k2 k − αk n Cλ, k/1 − α a k ≥0, 2.2 which proves univalence. Note that f is sense-preserving in U. This is because h z ≥ 1 − ∞ k2 k a k |z| k−1 > 1 − ∞ k2 k − αk n Cλ, k 1 − α a k ≥ ∞ k1 k αk n Cλ, k 1 − α b k > ∞ k1 k αk n Cλ, k 1 − α b k |z| k−1 ≥ ∞ k1 k b k |z| k−1 ≥ g z . 2.3 Using the fact that Rew>αif and only if |1 − α w|≥|1 α − w|,itsuffices to show that 1 − αD n λ fzD n1 λ fz − 1 αD n λ fz −D n1 λ fz ≥ 0. 2.4 4 Journal of Inequalities and Applications Substituting D n λ fz in 2.4 yields, by 2.1,weobtain 1 − αD n λ fzD n1 λ fz − 1 αD n λ fz −D n1 λ fz 2 − αz ∞ k2 k 1 − αk n Cλ, ka k z k − −1 n ∞ k1 k − 1 αk n Cλ, kb k z k − − αz ∞ k2 k − 1 − αk n Cλ, ka k z k − −1 n ∞ k1 k 1 αk n Cλ, kb k z k ≥21−α|z| 1− ∞ k2 k−αk n Cλ, k 1 − α a k |z| k−1 ∞ k1 kαk n Cλ, k 1 − α b k |z| k−1 ≥ 21 − α 1 − ∞ k2 k − αk n Cλ, k 1 − α a k − ∞ k1 k αk n Cλ, k 1 − α b k . 2.5 This last expression is nonnegative by 2.1, and so the proof is complete. The harmonic function fzz ∞ k2 1 − α k − αk n Cλ, k x k z k ∞ k1 1 − α k αk n Cλ, k y k z k , 2.6 where n, λ ∈ N 0 and ∞ k2 |x k | ∞ k1 |y k | 1 show that the coefficient bound given by 2.1 is sharp. The functions of the form 2.6 are in M H n, λ, α because ∞ k1 k − α 1 − α a k k α 1 − α b k k n Cλ, k1 ∞ k2 x k ∞ k1 y k 2. 2.7 In the following theorem, it is shown that the condition 2.1 is also necessary for functions f n h g n ,whereh and g n are of the form 1.4. Theorem 2.2. Let f n h g n be given by 1.4.Thenf n ∈ M H n, λ, α if and only if ∞ k1 k − α a k k α b k k n Cλ, k ≤ 21 − α, 2.8 where a 1 1,n,λ∈ N 0 ,Cλ, k kλ−1 λ , and 0 ≤ α<1. Proof. Since M H n, λ, α ⊂ M H n, λ, α, we only need to prove the “if and only if” part of the theorem. To this end, for functions f n of the form 1.4, we notice that the condition 1.3 is equivalent to Re 1 − αz − ∞ k2 k − αk n Cλ, ka k z k − −1 2n ∞ k1 k αk n Cλ, kb k z k z − ∞ k2 k n Cλ, ka k z k −1 2n ∞ k1 k n Cλ, kb k z k ≥ 0. 2.9 K. Al-Shaqsi and M. Darus 5 The above required condition 2.9 must hold for all values of z in U. Upon choosing the values of z on the positive real axis, where 0 ≤ z r<1, we must have 1 − α − ∞ k2 k − αk n Cλ, ka k r k−1 − ∞ k1 k αk n Cλ, kb k r k−1 1 − ∞ k2 k n Cλ, ka k r k−1 ∞ k1 k n Cλ, kb k r k−1 ≥ 0. 2.10 If the condition 2.8 does not hold, then the numerator in 2.10 is negative for r sufficiently close to 1. Hence there exist z 0 r 0 in 0, 1 for which the quotient in 2.8 is negative. This contradicts the required condition for f n ∈ M H n, λ, α and so the proof is complete. 3. Distortion bounds In this section, we will obtain distortion bounds for functions in M H n, λ, α. Theorem 3.1. Let f n ∈ M H n, λ, α.Thenfor|z| r<1, one has f n z ≤ 1 b 1 r 1 2 n λ 1 1 − α 2 − α − 1 α 2 − α b 1 r 2 , f n z ≥ 1 − b 1 r − 1 2 n λ 1 1 − α 2 − α − 1 α 2 − α b 1 r 2 . 3.1 Proof. We only prove the left-hand inequality. The proof for the right-hand inequality is similar and will be omitted. Let f n ∈ M H n, λ, α. Taking the absolute value of f n ,weobtain f n z z − ∞ k2 a k z k −1 n ∞ k1 b k z k ≥ 1 − b 1 r − ∞ k2 a k b k r k ≥ 1 − b 1 r − r 2 ∞ k2 a k b k ≥ 1− b 1 r− 1 − α 2 − α2 n λ 1 ∞ k2 2 − α2 n λ 1 1 − α a k 2 − α2 n λ 1 1 − α b k r 2 ≥ 1− b 1 r− 1 − α 2−α2 n λ1 ∞ k2 k−αk n Cλ, k 1 − α a k kαk n Cλ, k 1 − α b k r 2 ≥ 1 − b 1 r − 1 − α 2 − α2 n λ 1 1 − 1 α 1 − α b 1 r 2 . 3.2 The functions fzz b 1 z 1 2 n λ 1 1 − α 2 − α − 1 α 2 − α b 1 z 2 , fz 1 − b 1 z − 1 2 n λ 1 1 − α 2 − α − 1 α 2 − α b 1 z 2 3.3 for |b 1 |≤1 − α/1 α show that the bounds given in Theorem 3.1 are sharp. 6 Journal of Inequalities and Applications The following covering result follows from the left-hand inequality in Theorem 3.1. Corollary 3.2. If the function f n h g n ,whereh and g given by 1.4 are in M H n, λ, α,then w : |w| < 2 n1 λ 1 − 1 − 2 n λ 1 − 1 α 2 n λ 12 − α − 2 n1 λ 1 − 1 − 2 n λ 11 α 2 n λ 12 − α b 1 ⊂f n U. 3.4 4. Convolution, convex combination, and extreme points In this section, we show that the class M H n, λ, α is invariant under convolution and convex combination of its member. For harmonic functions f n zz − ∞ k2 a k z k −1 n ∞ k1 b k z k and F n zz − ∞ k2 A k z k −1 n ∞ k1 B k z k , the convolution of f n and F n is given by f n ∗F n zf n z∗F n zz − ∞ k2 a k A k z k −1 n ∞ k1 b k B k z k . 4.1 Theorem 4.1. For 0 ≤ β ≤ α<1,letf n ∈ M H n, λ, α and F n ∈ M H n, λ, β.Thenf n ∗F n ∈ M H n, λ, α ⊂ M H n, λ, β. Proof. We wish to show that the coefficients of f n ∗F n satisfy the required condition given in Theorem 2.2.ForF n ∈ M H n, λ, β, we note that |A k |≤1and|B k |≤1. Now, for the convolution function f n ∗F n ,weobtain ∞ k2 k − βk n Cλ, k 1 − β a k A k ∞ k1 k βk n Cλ, k 1 − β b k B k ≤ ∞ k2 k − βk n Cλ, k 1 − β a k ∞ k1 k βk n Cλ, k 1 − β b k ≤ ∞ k2 k − αk n Cλ, k 1 − α a k ∞ k1 k αk n Cλ, k 1 − α b k ≤ 1, 4.2 since 0 ≤ β ≤ α<1andf n ∈ M H n, λ, α.Thereforef n ∗F n ∈ M H n, λ, α ⊂ M H n, λ, β. We now examine the convex combination of M H n, λ, α. Let the functions f n j z be defined, for j 1, 2, ,by f n j zz − ∞ k2 a k,j z k −1 n ∞ k1 b k,j z k . 4.3 Theorem 4.2. Let the functions f n j z defined by 4.3 be in the class M H n, λ, α for every j 1, 2, ,m. Then the functions t j z defined by t j z m j1 c j f n j z, 0 ≤ c j ≤ 1 4.4 are also in the class M H n, λ, α,where m j1 c j 1. K. Al-Shaqsi and M. Darus 7 Proof. According to the definition of t j ,wecanwrite t j zz − ∞ k2 m j1 c j a k,j z k −1 n ∞ k1 m j1 c j b n,j z k . 4.5 Further, since f n j z are in M H n, λ, α for every j 1, 2, ,thenby2.8,wehave ∞ k1 k − α m j1 c j a k,j k α m j1 c j b k,j k n Cλ, k m j1 c j ∞ k1 k − α a n,j k α b n,j k n Cλ, k ≤ m j1 c j 21 − α ≤ 21 − α. 4.6 Hence the theorem follows. Corollary 4.3. The class M H n, λ, α is closed under convex linear combination. Proof. Let the functions f n j zj 1, 2 defined by 4.1 be in the class M H n, λ, α. Then the function Ψz defined by Ψzμf n 1 z1 − μf n 2 z, 0 ≤ μ ≤ 1 4.7 is in the class M H n, λ, α. Also, by taking m 2, t 1 μ,andt 2 1 − μ in Theorem 4.1,we have the corollary. Next we determine the extreme points of closed convex hulls of M H n, λ, α denoted by clcoM H n, λ, α. Theorem 4.4. Let f n be given by 1.4.Thenf n ∈ M H n, λ, α if and only if f n z ∞ k1 X k h k zY k g n k z , 4.8 where h 1 zz, h k zz − 1 − α/k − αk n Cλ, kz k ,k 2, 3, , g n k zz −1 n 1 − α/k αk n Cλ, kz k ,k 1, 2, 3, , and ∞ k1 X k Y k 1,X k ≥ 0,Y k ≥ 0. In particular, the extreme points of M H n, λ, α are h k and g n k . Proof. For the functions f n of the form 4.8,wehave f n z ∞ k1 X k h k zY k g n k z ∞ k1 X k Y k z − ∞ k2 1 − α k − αk n Cλ, k X k z k −1 n ∞ k1 1 − α k αk n Cλ, k Y k z k . 4.9 Then ∞ k2 k − αk n Cλ, k 1 − α a k ∞ k1 k αk n Cλ, k 1 − α b k ∞ k2 X k ∞ k1 Y k 1 − X 1 ≤ 1, 4.10 8 Journal of Inequalities and Applications and so f n ∈ clcoM H n, λ, α. Conversely, suppose that f n ∈ clcoM H n, λ, α. Setting X k k − αk n Cλ, k 1 − α a k , 0 ≤ X k ≤ 1,k 2, 3, , Y k k αk n Cλ, k 1 − α b k , 0 ≤ Y k ≤ 1,k 1, 2, 3, , 4.11 and X 1 1 − ∞ k2 X k − ∞ k1 Y k . Therefore, f n can be written as f n zz − ∞ k2 a k z k −1 n ∞ k1 b k z k z − ∞ k2 1 − αX k k − αk n Cλ, k z k −1 n ∞ k1 1 − αY k k αk n Cλ, k z k z ∞ k2 h k z − z X k ∞ k1 g n k z − z Y k ∞ k2 h k zX k ∞ k1 g n k zY k z 1 − ∞ k2 X k − ∞ k1 Y k ∞ k1 h k zX k g n k zY k , as required. 4.12 Using Corollary 4.3 we have clcoM H n, λ, αM H n, λ, α. Then the statement of Theorem 4.4 is really for f ∈ M H n, λ, α. 5. An application of neighborhood In this section, we will prove that the functions in a neighborhood of M H n, λ, α are starlike harmonic functions. Following 10, we defined the δ-neighborhood of a function f ∈TH by N δ f Fzz − ∞ k2 A k z k − ∞ k1 B k z k , ∞ k2 k a k − A k b k − B k b 1 − B 1 ≤ δ , 5.1 where δ>0. Theorem 5.1. Let δ 2 − α2 n λ 1 − 1 α − 2 − α2 n λ 1 − 1 − α b 1 2 − α2 n λ 1 . 5.2 Then N δ M H n, λ, α ⊂TH. K. Al-Shaqsi and M. Darus 9 Proof. Suppose f n ∈ M H n, λ, α.LetF n H G n ∈N δ f n , where H z − ∞ k2 A k z k and G n −1 n ∞ k1 B k z k . We n eed to show that F n ∈TH. In other words, it suffices to show that F n satisfies the condition TF ∞ k2 k|A k | |B k ||B 1 |≤1. We observe that TF ∞ k2 k A k B k B 1 ∞ k2 k A k − a k a k B k − b k b k B 1 − b 1 b 1 ∞ k2 k A k − a k B k − b k ∞ k2 k a k b k B 1 − b 1 b 1 ∞ k2 k A k − a k B k − b k B 1 − b 1 ∞ k2 k a k b k b 1 δ b 1 ∞ k2 k a k b k δ b 1 1 − α 2 − α2 n λ 1 ∞ k2 2 − α 1 − α a k 2 α 1 − α b k 2 n λ 1 ≤ δ b 1 1 − α 2 − α2 n λ 1 ∞ k2 k − α 1 − α a k k α 1 − α b k k n Cλ, k ≤ δ b 1 1 − α 2 − α2 n λ 1 1 − 1 α 1 − α b 1 . 5.3 Now this last expression is never greater than one if δ ≤ 1 − b 1 − 1 − α 2 − α2 n λ 1 1 − 1 α 1 − α b 1 2 − α2 n λ 1 − 1 α − 2 − α2 n λ 1 − 1 − α b 1 2 − α2 n λ 1 . 5.4 Acknowledgment The w ork presented here was supported by Fundamental Research Grant Scheme UKM-ST-01- FRGS0055-2006. References 1 K. Al-Shaqsi and M. Darus, “An operator defined by convolution involving the polylogarithms func- tions,” submitted. 2 J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae. Series A I, vol. 9, pp. 3–25, 1984. 3 J. M. Jahangiri, “Harmonic functions starlike in the unit disk,” Journal of Mathematical Analysis and Applications, vol. 235, no. 2, pp. 470–477, 1999. 4 J. M. Jahangiri, G. Murugusundaramoorthy, and K. Vijaya, “Salagean-type harmonic univalent func- tions,” Southwest Journal of Pure and Applied Mathematics, no. 2, pp. 77–82, 2002. 10 Journal of Inequalities and Applications 5 G. Murugusundaramoorthy and K. Vijaya, “On certain classes of harmonic functions involving Ruscheweyh derivatives,” Bulletin of the Calcutta Mathematical Society, vol. 96, no. 2, pp. 99–108, 2004. 6 H. Silverman, “Harmonic univalent functions with negative coefficients,” Journal of Mathematical Anal- ysis and Applications, vol. 220, no. 1, pp. 283–289, 1998. 7 H. Silverman and E. M. Silvia, “Subclasses of harmonic univalent functions,” New Zealand Journal of Mathematics, vol. 28, no. 2, pp. 275–284, 1999. 8 J. M. Jahangiri, “Coefficient bounds and univalence criteria for harmonic functions with negative co- efficients,” Annales Universitatis Mariae Curie-Skłodowska. Sectio A, vol. 52, no. 2, pp. 57–66, 1998. 9 J. M. Jahangiri and H. Silverman, “Harmonic univalent functions with varying arguments,” Interna- tional Journal of Applied Mathematics, vol. 8, no. 3, pp. 267–275, 2002. 10 S. Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathematical So- ciety, vol. 81, no. 4, pp. 521–527, 1981. . Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 263413, 10 pages doi:10.1155/2008/263413 Research Article On Harmonic Functions Defined by Derivative Operator K α b 1 ⊂f n U. 3.4 4. Convolution, convex combination, and extreme points In this section, we show that the class M H n, λ, α is invariant under convolution and convex combination of its member. For harmonic functions. class of Salagean-type harmonic univalent functions introduced by Jahangiri et al. 4;andM H 0,λ,α is the class of Ruscheweyh-type harmonic univalent functions stud- ied by Murugusundaramoorthy
Ngày đăng: 22/06/2014, 06:20
Xem thêm: Báo cáo hóa học: " Research Article On Harmonic Functions Defined by Derivative Operator" docx, Báo cáo hóa học: " Research Article On Harmonic Functions Defined by Derivative Operator" docx
