ON A SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS DEFINED BY CONVOLUTION. G. Shelake, S. Joshi, S. Halim

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1 Acta Uiversitatis Apulesis ISSN: No. 38/2014 pp ON A SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS DEFINED BY CONVOLUTION G. Shelake, S. Joshi, S. Halim Abstract. I this paper, we itroduce ad study a subclass of harmoic uivalet fuctios defied by covolutio ad itegral covolutio. Coefficiet bouds, extreme poits, distortio bouds, covolutio coditios ad covex combiatio are determied for fuctios i this family Mathematics Subject Classificatio: 30C45, 30C50. Keywords: covolutio. Harmoic uivalet fuctios, extreme poits, distortio bouds, 1. Itroductio A cotiuous fuctio f is said to be a complex-valued harmoic fuctio i a simply coected domai D i complex plae C if both real part of f ad imagiary part of f are real harmoic i D. Such fuctios ca be expressed as f = h + ḡ 1 where h ad g are aalytic i D. We call h the aalytic part ad g the co-aalytic part of f. A ecessary ad sufficiet coditio for f to be locally uivalet ad sese-preservig i D is that h > g for all i D, see [3]. Every harmoic fuctio f = h + ḡ is uiquely determied by the coefficiets of power series expasios i the uit disk U = : < 1} give by h = + a, g = b, U, b 1 < 1, 2 where a C for = 2, 3, 4,... ad b C for = 1, 2, 3,.... For further iformatio about these mappigs, oe may refer to [1, 3, 5, 8, 10, 11]. I 1984, Cluie ad Sheil-Small [3] studied the family S H of all uivalet sesepreservig harmoic fuctios f of the form 1 i U, such that h ad g are represeted by 2. Note that S H reduces to the well-kow family S, the class of all 251

2 ormalied aalytic uivalet fuctios h give i 2, wheever the co-aalytic part g of f is ero. Let K ad K H deote the respective subclasses of S ad S H where the images of fu are covex. The covolutio of two fuctios of the form is give by Φ = + µ ad Ψ = + ν, µ, ν 0 3 Φ Ψ = Φ Ψ = + ad the itegral covolutio is defied by Φ Ψ = Φ Ψ = + µ ν µ ν Towards the ed of last cetury, Jahagiri [8], Frasi [7], Silverma [10], ad Silverma ad Silvia [11] were amogst those who focused o the harmoic starlike fuctios. Later Oturk S. et. al [9] defied the class SH λ, α cosistig of fuctios f = h + ḡ such that h ad g are of the form h = which satisfy the coditio Re a, g = h g b 4 λh g + 1 λh + g } α, for some 0 α < 1, 0 λ 1 ad for all U. Let S H Φ, Ψ, λ, α deote the subclass of S H of fuctios of the form f = h+ḡ S H that satisfy the coditio Re h Φ g Ψ λh Φ g Ψ + 1 λh Φ + g Ψ } α, 5 where 0 α < 1, 0 λ 1 ad Φ, Ψ are as give i 3. We further let T SH Φ, Ψ, λ, α deote the subclass of S H Φ, Ψ, λ, α cosistig of fuctios f = h + ḡ S H such that h ad g are of the form 4. We ote that the family 252

3 T SH Φ, Ψ, λ, α is of special iterest because it cotais various classes of wellkow harmoic uivalet fuctios as well as may ew oes. For differet choice of Φ, Ψ, λ ad α we obtai followig various classes itroduced by other authors: i T S H ii T S H iii T S H iv T S H, 1 2, 1 2, 1 2, 1 2, λ, α = T S 1 2 H λ, α see Oturk et al.[9]., 0, α = T S 1 2 H α see Jahagiri [8]. 1 2, 0, 0 = T S H see Silverma et al. [11]., 0, 0 = T S H see Avci et al.[2] ad Silverma [10]. v T SH +2 +, 2, 0, α = K H α see Jahagiri [8]. vi T SH +2 +, 2, 0, 0 = K H see Silverma [10]. vii T SH Φ, Ψ, 0, α = T S H Φ, Ψ, α see Dixit et al.[4]. viii T SH Φ, Ψ, 0, α = HST φ, χ, 0, α see El-Ashwah[6] ad Dixit et al.[4] I this paper, we obtai coefficiet bouds for the subclasses SH Φ, Ψ, λ, α ad T SH Φ, Ψ, λ, α, we also obtai distortio bouds, extreme poits, covolutio coditios, ad covex combiatio for fuctios i T SH Φ, Ψ, λ, α. 2. Mai Results We begi with a sufficiet coditio for fuctios i SH Φ, Ψ, λ, α. Theorem 1. Let f = h + ḡ be of the form 2. Furthermore, let µ α αλ 1 a + ν + α αλ + 1 b 1, 6 where 0 α < 1, 0 λ 1, 2 µ [ 1 + αλ λ + 1] ν [ 1 + αλ + λ 1]. The f is sese-preservig harmoic uivalet i U ad for λ 1 α 1+α, f S H Φ, Ψ, λ, α. 253

4 Proof. We first ote that f is sese-preservig i U. This is because h 1 a 1 > 1 a 1 ν µ α αλ 1 + α αλ + 1 b 1 g, a b > ν + α αλ + 1 b 1 where we have used hypothesis of the theorem. Now to show that f is uivalet i U, suppose 1, 2 U so that 1 2, the f 1 f 2 h 1 h 2 1 g 1 g 2 h 1 h 2 b 1 2 = a 1 2 b > 1 1 a ν +α αλ+1 1 α µ α αλ 1 1 α b a Now, we show that f SH Φ, Ψ, λ, α. By usig the fact that Rew > α if ad oly if + w > 1 + α w, it suffices to show that, B + A 1 + αb A > 0, 7 where A = h Φ g Ψ ad B = λa + 1 λh Φ + g Ψ. 254

5 Substitutig A ad B i 7 as well as makig use of 6 ad λ 1 α 1+α, we obtai A + B A 1 + αb = [1 + λ]h Φ g Ψ + 1 λh Φ + g Ψ [1 λ1 + α]h Φ g Ψ 1 + α1 λh Φ + g Ψ = 2 α + [ 1 + λ + 1 λ ] µ a [ 1 + λ 1 λ ] ν b where ν 1 = 1 α + [ α [ α λ 1 λ λ + 1 λ ] µ a ] ν b where ν 1 = 1 α αλ µ a 1 } + α αλ + 1 ν b 1 α αλ 1 > 2 1 µ a 0 from 6. + α αλ + 1 ν b The coefficiet boud 6 is sharp for the fuctios f = + where µ } x + α αλ 1 x + y = 1. ν y, + α αλ

6 Next, we show that the above sufficiet coditio is also ecessary for fuctios i T SH Φ, Ψ, λ, α. Theorem 2. Let f = h + ḡ be of the form 4. The f T SH Φ, Ψ, λ, α if ad oly if µ α αλ 1 a + ν + α αλ + 1 b 1, 8 where, 0 α < 1, 0 λ 1, 2 µ [ 1 + αλ λ + 1] ν [ 1 + αλ + λ 1]. Proof. The if part, follows from Theorem 1. To prove the oly if part, let f T SH Φ, Ψ, λ, α the from 5 we have } h Φ g Ψ Re λh Φ g Ψ + 1 λh Φ + g Ψ α = Re > 0. [ α αλ 1] µ a µ [ λ + 1 λ ] a + If we choose to be real ad 1, we get or, equivaletly, 1 [ α αλ 1] µ a µ [ λ + 1 λ µ [ α αλ 1] ] a + a + which is the required coditio 8. ν [+α αλ+1] ν [ 1 λ λ ] b ν [+α αλ+1] b ν [ 1 λ λ ] b b 0, [ + α αλ + 1] ν b, I additio to the above mai result, the followig results are further properties cocerig the class T SH Φ, Ψ, λ, α. These results agree with previously obtaied oes by other authors. 256

7 Theorem 3. If f T SH Φ, Ψ, λ, α ad µ 22 α αλ µ α αλ 1 ν + α αλ + 1 for 2. The we have, f 1 + b 1 r + 2 µ 2 2 α αλ 1 + α 2αλ µ 2 2 α αλ ν 1 b 1 r 2, = r < 1, ad f 1 b 1 r 2 µ 2 2 α αλ 1 + α 2αλ µ 2 2 α αλ ν 1 b 1 r 2, = r < 1, Proof. We oly prove the right had iequality. The proof for the left had iequality is similar ad will be omitted. f 1 + b 1 r b 1 r b 1 r + a + b r a + b r 2 2 µ 2 2 α αλ µ 2 2 α αλ a + b r b 1 r + µ 1 + b 1 r + 2 µ 2 2 α αλ α αλ 1 a + ν + α αλ + 1 b r α 2αλ ν 1 b 1 µ 2 2 α αλ 1 + b 1 r + 2 µ 2 2 α αλ 1 + α 2αλ µ 2 2 α αλ ν 1 b 1 r 2. r 2 The upper boud give for f T SH Φ, Ψ, λ, α is sharp ad equality occurs for the fuctio f = + b 1 +2 µ 2 2 α αλ 1 + α 2αλ µ 2 2 α αλ ν 1 b 1 2, b α 2αλν

8 3. The followig coverig result follows from the left had iequality i Theorem Corollary 4. Let f T SH Φ, Ψ, λ, α, the w : w < 1 } A [A α 2αλν 1 A b 1 ] fu, where A = µ α αλ. Now we determie the extreme poits of T SH Φ, Ψ, λ, α Theorem 5. Let h 1 =, h = = 2, 3,... µ α αλ 1 ad g = + = 1, 2... ν + α αλ + 1 The f T SH Φ, Ψ, λ, α if ad oly if it ca be expressed as f = x h + y g, where x 0, y 0, x 1 = 1 x + y 0, ad y 1 = 0. I particular, the extreme poits of T S H Φ, Ψ, λ, α are h ad g. Proof. Suppose f = x h + y g = x + y µ α αλ 1 + y. + α αλ + 1 ν x The µ α αλ 1 µ α αλ 1 x 258

9 ν + α αλ = ν x + y = 1 x α αλ + 1 ad so f T SH Φ, Ψ, λ, α. Coversely, if f T S H Φ, Ψ, λ, α, the a ad b ν µ α αλ 1 Settig ad x = µ y = ν α αλ 1 + α αλ + 1 y + α αλ + 1 a = 2, 3... b = 1, 2... The ote that by Theorem 2, 0 x 1 = 2, 3... ad 0 y 1 = 1, 2... We defie x 1 = 1 x y, by Theorem 2 we obtai f = x h + y g. This completes the proof of Theorem 5. Next, we show that T SH Φ, Ψ, λ, α is closed uder covex combiatios of its members. Theorem 6. The class T SH Φ, Ψ, λ, α is closed uder covex combiatio. Proof. For i = 1, 2, 3... let f i T S H Φ, Ψ, λ, α, where f i is give by The by Theorem 2, µ α αλ 1 f i = a i + b i. a i + ν + α αλ + 1. b i 1 9 For t i = 1, 0 t i 1, the covex combiatio of f i may be writte as t i f i = t i a i + t i b i. 259

10 The by 6, µ α αλ 1 t i a i + ν + α αλ + 1 t i b i = 1 t i µ t i = 1. from 9 α αλ 1 ad so by Theorem 2, we have a i + ν + α αλ + 1 t i f i T SH Φ, Ψ, λ, α. } b i Fially we show that the class T SH Φ, Ψ, λ, α is ivariat uder covolutio. For harmoic fuctios f = a + b ad F = A + B, we defie the covolutio of two harmoic fuctios f ad F as f F = f F = a A + b B. Theorem 7. If f T SH Φ, Ψ, λ, α ad F T S H Φ, Ψ, λ, α the f F T SH Φ, Ψ, λ, α. Proof. Let f = a + b ad F = A + B be i T SH Φ, Ψ, λ, α, The by Theorem 2, we have ad µ µ α αλ 1 α αλ 1 a + A + ν ν + α αλ α αλ + 1 b 1, B

11 So for the coefficiets of f F we ca write µ α αλ 1 a A + ν + α αλ + 1 b B µ α αλ 1 a + ν + α αλ + 1 b 1. Thus f F T SH Φ, Ψ, λ, α. Ackowledgemets. The first author is thakful to Uiversity Grat Commissio, New Delhi, for fiacial support vide grat No /12WRO. Refereces [1] O. Ahuja, Plaar harmoic uivalet ad related mappigs, J. Ieq Pure Appl. Math. 6, , Art 122. [2] Y. Avci, E. Zlotkiewic, O harmoic uivalet mappigs, A. Uiv.Mariae Curie Sklodowska Sect. A , 1-7. [3] J. Cluie, T. Sheil-Small, Harmoic uivalet fuctios, A. Acad Sci. Fe. Ser. A. I. Mat , [4] k. Dixit, A. Pathak, S. Porwal, R. Agarwal, O a subclass of harmoic uivalet fuctios defied by covolutio ad itegral covolutio, Iteratioal Joural of Pure ad Applied Mathematics 69, , [5] P. Dure, Harmoic mappigs i the plae, Cambridge Tract i Mathematics Cambridge Uiversity Press2004, ISBN [6] El-Ashwah, Subclass of uivalet harmoic fuctios defied by dual covolutio, Joural of Iequalities ad Applicatios 2013, 2013:537. [7] B. Frasi, Comprehesive family of harmoic uivalet fuctios SUT J. Math.42, , [8] J. Jahagiri, Harmoic fuctios starlike i the uit disc, J. Math. Aal.Appl , [9] M. Oturk, S. Yalci, M. Yamakaradei, Covex subclasses of harmoic starlike fuctios, Applied Mathematics ad Computatio , [10] H. Silverma, Harmoic uivalet fuctios with egative coefficiets, J. Math Aal Appl ,

12 [11] H. Silverma, E. Silvia, Subclasses of harmoic uivalet fuctios, New Zealad J , Girish D. Shelake Departmet of Mathematics, Willigdo College, Sagli Maharashtra,Idia Satosh B. Joshi Departmet of Mathematics, Walchad College, Sagli , Maharashtra,Idia Sueii A. Halim Istitute of Mathematical scieces, Uiversit Malaya Malaysia 262