In this paper, we obtain some sufficient conditions for generalized Sakaguchi type function of order β, defined on the open unit disk. Many interesting outcomes of our results are also calculated.Publisher's Versio

Mathur, A. Trilok

Mathur, Ruchi

Sinha, C. Deepa

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. Eng. Math. V.4, N.1, 2014, pp. 127-131.SUFFICIENT CONDITIONS FOR GENERALIZED SAKAGUCHI TYPEFUNCTIONS OF ORDER βA. TRILOK MATHUR1, B. RUCHI MATHUR2, C. DEEPA SINHA3 §Abstract. In this paper, we obtain some sufficient conditions for generalized Sakaguchitype function of order β, defined on the open unit disk. Many interesting outcomes ofour results are also calculated.Keywords: Generalized Sakaguchi type function of order β, Univalent functions.AMS Subject Classification: 30C45 , 30C50 , 30C80.1. IntroductionLet An be the class of the formf(z) = z + an+1zn+1 + .... (1)that are analytic in the unit disk ∆ = {z ∈ C : |z| < 1} and let A1 = A. An analyticfunction f(z) ∈ An is said to be in the generalized Sakaguchi class Sn(β, s, t) if it satisfiesRe{(s− t)zf ′(z)f(sz)− f(tz)}> β, z ∈ ∆ (2)for some β(0 ≤ β < 1), s and t are real parameters, s > t and for all z ∈ ∆.For n = 1 the generalized Sakaguchi class Sn(β, s, t) reduces to the subclass S(β, s, t)studied by Frasin [[2], see also [6], [7]]. For n = 1, s = 1, this class is reduced to S(β, t)studied by Owa et al. [9, 10], Goyal and Goswami [3] and Cho et al.[1]. The class S(0,−1)was introduced by Sakaguchi [12]. Recently T. Mathur et al. [[6], [7]] have introduced andstudied some properties of S(β, s, t) .In this paper, we obtain some sufficient conditions for functions f(z) ∈ Sn(β, s, t). Toprove our results, we need the following:Lemma 1.1 (8). Let Ω be a set in the complex plane C and suppose that ϕ is a mappingfrom C2 × ∆ to C which satisfies ϕ(ix, y; z) ̸∈ Ω for z ∈ ∆, and for all real x, y suchthat y ≤ −n(1 + x2)/2. If the function p(z) = 1 + cnzn + ...... is analytic in ∆ andϕ(p(z), zp′(z); z) ∈ Ω for all z ∈ ∆, then Re(p(z)) > 0.1 Birla Institute of Technology and Science, Pilani, India ,e-mail: tmathur@pilani.bits-pilani.ac.in2 Jaipur Engineering College and Research Centre, Jaipur, India,e-mail: ruchs 21@yahoo.co.in3 South Asian University, New Delhi, India,e-mail: deepasinha2001@gmail.com§ Manuscript received September 13, 2013; revised: November 21, 2013.TWMS Journal of Applied and Engineering Mathematics Vol.4 No.2 c⃝ Işık University, Departmentof Mathematics 2014; all rights reserved.12 TWMS J. APP. ENG. MATH. V.4, N.1, 20142. Main ResultsTheorem 2.1. If f(z) ∈ An satisfiesRe[(s− t)2zf ′(sz)f(sz)− f(tz){αszf ′′(sz)f ′(sz)+αtzf ′(tz)f(sz)− f(tz)+ 1}]> αβ{sβ +n2(s− t)− (s− t)}+{β − nα2}(s− t) (3)for (z ∈ ∆, 0 ≤ α ≤ 1, 0 ≤ β < 1 and t < s), then f(z) ∈ Sn(β, s, t).Proof. Define p(z) by {(s− t)zf ′(sz)f(sz)− f(tz)}= (1− β)p(z) + β.Then p(z) = 1 + cnzn + ..... and is analytic in ∆.A computation shows thatszf ′′(sz)f ′(sz)+tzf ′(tz)f(sz)− f(tz)=(s− t)(1− β)zp′(z) + s[(1− β)p(z) + β]2 − (s− t)[(1− β)p(z) + β](s− t)[(1− β)p(z) + β]and hence(s− t)2zf ′(sz)f(sz)− f(tz)[αszf ′′(sz)f ′(sz)+αtzf ′(tz)f(sz)− f(tz)+ 1]= α(s−t)(1−β)zp′(z)+αs(1−β)2p2(z)+(1−β[2sαβ+(s−t)(1−α)]p(z)+β[sαβ+(s−t)(1−α)](4)= ϕ(p(z), zp′(z); z) (say)whereϕ(u, v; z) = α(s−t)(1−β)v+αs(1−β)2u2+(1−β)[2sαβ+(s−t)(1−α)]u+β[sαβ+(s−t)(1−α)]For all real x and y satisfying y ≤ −n(1 + x2)/2, we haveRe[ϕ(ix, y; z)] ≤ α(s− t)(1− β)y − αs(1− β)2x2 + β[sαβ + (s− t)(1− α)]≤ α(s− t)(1− β){−(1 + x2)2}− αs(1− β)2x2 + β[sαβ + (s− t)(1− α)]=−αn2(s− t)(1− β)−{αn2(s− t)(1− β) + αβ(1− β)2}x2 + β[sαβ + (1− α)(s− t)]≤ −αn2(s− t)(1− β) + β[sαβ + (1− α)(s− t)]= αβ{sβ +n2(s− t)− (s− t)}+{β − nα2}(s− t)Let Ω ={w;Re(w) > αβ{β + n2 (s− t)− (s− t)}+{β − nα2}(s− t)}Then ϕ(p(z), zp′(z); z) ∈ Ω and ϕ(ix, y; z) ̸∈ Ω for all real x and y ≤ −n(1+x2)/2, z ∈ ∆.By an application of Lemma 1.1, the result follows. Remark 2.1. On putting s = 1, in Theorem 2.1, we get the known results due to Goyalet al.[9]T. MATHUR, R. MATHUR , D. SINHA : SUFFICIENT CONDITIONS FOR GENERALIZED ... 3Theorem 2.2. Let 0 ≤ β < 1, t < s with − 1 ≤ ts + β < 1,λ = (1− β)2{n2(s− t) + s(1− β}2, µ ={n2|(s− t)|(1− β) + β|(s− t− sβ)|}2,ν ={s(1− β)2 − β(s− t− sβ)}2and σ = {(1− β)(2sβ − t− s)}2 (5)satisfy(λ+ µ− ν + σ)β2 < (1− 2β)µ.Also suppose that r0 be the positive real root of the equation2λ(1− β)2r3 +{(1− β)2(2λ+ µ− ν + σ) + 3λβ2}r2 + 2β2(2λ+ µ− ν + σ)r+(λ+ 2µ− ν + σ)β2 − (1− β)2µ = 0 (6)andρ2 =(1− β)2(1 + r0)(s− t)2 {(1− β)2r0 + β2}[λr20 + (λ+ µ− ν + σ)r0 + µ] (7)Now if f(z) ∈ An satisfies∣∣∣∣( s− t)zf ′(sz)f(sz)− f(tz) − 1)(szf ′′(sz)f ′(sz)+tzf ′(tz)f(sz)− f(tz))∣∣∣∣ ≤ ρ (z ∈ ∆)then f(z) ∈ Sn(β, s, t).Proof. Define p(z) by {(s− t)zf ′(sz)f(sz)− f(tz)}= (1− β)p(z) + β.Then p(z) = 1 + cnzn + ..... and is analytic in ∆.A computation shows thatszf ′′(sz)f ′(sz)+tzf ′(tz)f(sz)− f(tz)=(s− t)(1− β)zp′(z) + s[(1− β)p(z) + β]2 − (s− t)[(1− β)p(z) + β](s− t)[(1− β)p(z) + β]and hence ((s− t)zf ′(sz)f(sz)− f(tz)− 1)(szf ′′(sz)f ′(sz)+tzf ′(tz)f(sz)− f(tz))=(1− β)(p(z)− 1)(s− t)[(1− β)p(z) + β]{(s− t)(1− β)zp′(z) + s[(1− β)p(z) + β]2 − (s− t)[(1− β)p(z) + β]}= ϕ(p(z), zp′(z); z)Then for all real x and y satisfying y ≤ −n(1 + x2)/2, we have|ϕ(ix, y; z)|2 = (1− β)2(1 + x2)(s− t)2[(1− β)2x2 + β2]×[(s− t)(1− β)y − s(1− β)2x2 − β(s− t− sβ)]2+ (1− β)2[2sβ − (s− t)]2x2=(1− β)2(1 + r)(s− t)2[(1− β)2r + β2]×[(s− t)(1− β)y − s(1− β)2r − β(s− t− sβ)]2+ (1− β)2[2sβ − (s− t)]2r= g(r, y)where r = x2 > 0 and y ≤ −n(1 + x2)/2Since∂g∂y=2(1− β)3(1 + r)(s− t)[(1− β)2r + β2]{(s− t)(1− β)y − β(s− t− sβ)− s(1− β)2r}< 0therefore we haveh(r) = g [r,−n(1 + r)/2] ≤ g(r, y),4 TWMS J. APP. ENG. MATH. V.4, N.1, 2014whereh(r) =(1− β)2(1 + r)(s− t)2[(1− β)2r + β2][λr2 + (λ+ µ− ν + σ)r + µ] (8)where λ, µ, ν, and σ are given in (5).Now differentiating (8) and using h′(r) = 0, we get2λ(1− β)2r3 +{(1− β)2(2λ+ µ− ν + σ) + 3λβ2}r2+2β2(2λ+ µ− ν + σ)r + (λ+ 2µ− ν + σ)β2 − (1− β)2µ = 0which is a cubic equation in r. Since r0 is the positive real root of this equation we haveh(r) ≥ h(r0) and hence|ϕ(ix, y; z)|2 ≥ h(r0) = ρ2.Define Ω = {w; |w| < ρ}, then ϕ(p(z), zp′(z); z) ∈ Ω for all real x andy ≤ −n(1+x2)/2, z ∈ ∆. Therefore by an application of Lemma 1.1 the result follows. Remark 2.2. By taking s = 1 in Theorem 2.2 we get the known results of Goyal et al.[4]For s = 1 and t = 0 in Theorem 2.2 gives the known results due to Ravichandran et al.[11]and for n = 1, β = 0, t = 0, our Theorem 2.2 reduces to another known result of Li andOwa.[5]ACKNOWLEDGMENTAuthors are thankful to Prof. S.P. Goyal, Emeritus Scientist(CSIR), University of Ra-jasthan, Jaipur, India, for his kind help and valuable suggestions during the preparationof this paper.References[1] Cho, N. E., Kwon, O. S. and Owa, S., Certain Subclasses of Sakaguchi Functions., Southeast AsianBull. Math., 17 (1993), 121-126.[2] Frasin, B. A., Coefficient Inequalities for Certain Classes of Sakaguchi Type Functions, InternationalJ. Nonlinear Science, 10 (2010),(2), 206-211.[3] Goyal, S. P. and Goswami, P., Certain Coefficient Inequalities for Sakaguchi Type Functions andApplications to Fractional Derivative Operator., Acta universitaties Apulenses , 19(3) (2011), 159-166.[4] Goyal, S. P., Vijayvargiya, P. and Goswami, P., Sufficient Conditions For Sakaguchi Type Functionsof order β, European J. Pure Appl. Math. , 4 (2009), 230-236.[5] Li, J. L. and Owa, S., Sufficient Conditions for Starlikeness, Indian J. Pure Appl. Math. ,33 (2002),313-318.[6] Mathur, T. and Mathur. R., Some Starlike and Convexity Properties of Sakaguchi Classes for Hyper-geometric Functions, Int. J. Open Problems in Complex Analysis , 4 (1)(2012).[7] Mathur, T. and Mathur, R., Fekete- Szego Inequalities for Generalized Sakaguchi Type Functions,Proceeding of the World Congress on Engineering , 1(2012), 210-213.[8] Miller, S. S. and Mocanu, P. T., Differential Subordinations and Inequalities in the Complex Plane,J. Differ. Equations , 167 (1987), 199-211.[9] Owa, S., Sekine, T. and Yamakawa, R., On Sakaguchi Type Functions, Applied Mathematics andComputation, 187 (2007), 356-361.[10] Owa, S., Sekine, T. and Yamakawa, R., Notes on Sakaguchi Functions, RIMS Kokyuroku , 1414(2005), 76-82.[11] Ravichandran, V., Selvaraj, C. and Rajalakshmi, R., Sufficient Conditions for Starlike Functions ofOrder α, J. Inequal. Pure Appl. Math. , 3 (5)(2002), 1-6.[12] Sakaguchi, K., On a Certain Univalent Mapping, J. Math. Soc. Japan, 11 (1959), 72-75.

Sufficient conditions for generalized Sakaguchi type functions of order β

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Sufficient conditions for generalized Sakaguchi type functions of order β

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