Bulletin of the Transilvania University of Braşov Vol 8(57), No Series III: Mathematics, Informatics, Physics, 1-12

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1 Bulletin of the Transilvania University of Braşov Vol 857), No Series III: Mathematics, Informatics, Physics, 1-12 DISTORTION BOUNDS FOR A NEW SUBCLASS OF ANALYTIC FUNCTIONS AND THEIR PARTIAL SUMS Ibrahim AKTAŞ 1 and Halit ORHAN 2 Abstract In the resent aer, we introduce a new subclass of functions which are analytic in the oen unit disk. Also, we obtain coefficient inequalities for functions belonging to this class. Futhermore, we give some results associated with distortions bounds. In addition to that, we investigate lower bounds for artial sums of functions belonging to this class Mathematics Subject Classification: 30C45. Key words: valent function, starlike function, distortion bounds, artial sums. 1 Introduction Let A denote the class of functions fz) in the form fz) = z + a k+ z k+ 1) which are analytic and valent in the oen unit disk = {z : z C, z < 1} and N = {1, 2, 3,...}. Further, by A and S α), we denote the following classes: { } A = fz) = z + a k+1 z k+1 : f is analytic in and S α) = { zf ) } z) f A : Re > α, 0 α < 1, z fz) 1 Corresonding author, Deartment of Mathematical Engineering, Faculty of Engineering and Natural Science, Gümüşhane University, Gümüşhane-Turkey, aktasibrahim38@gmail.com 2 Deartment of Mathematics, Faculty of Science, Ataturk University, Erzurum-Turkey, e- mail: orhanhalit607@gmail.com

2 2 Ibrahim Aktaş and Halit Orhan resectively. We know that, S α) is a familiar subclass of A consisting of functions which are starlike of order α in. For the function classes S α) was given coefficient inequality by Silverman [6] in In 1991, Altıntas [1] gave coefficient inequality for a subclass of certain starlike functions with negative coefficients. Owa, Ochiai and Srivastava [4] introduced the subclass Mα) of the class A and they roved some theorems relations with coeficient inequality for this class in Kamali [7] defined a new subclass Mα, λ, Ω) of the class A and he investigated some roerties for this subclass in For the functions fz) belonging to the class A, we define the oerator D as follows: D 0 fz)) = fz) D 1 fz)) = Dfz)) = z ) f k + z) = z + a k+ z k+ ) k + 2 D 2 fz)) = DD 1 fz))) = z + a k+z k+. D Ω fz)) = DD Ω 1 fz))) = z + ) k + Ω a k+z k+. We note that, for = 1 we obtain Salagean differential oerator which was defined by Salagean [5]. In this work, we introduce a new subclass M α, λ, Ω) of the class A consisting of functions fz) such that 1 λ)d Ω fz)) + λd Ω+1 fz)) 1 λ) z DΩ fz))) + λ z + λ DΩ+1 fz))) 2α < + λ 2α z, 0 < α < 1, 0 λ < 1, Ω N 0 = {0, 1, 2, 3,...}, N = {1, 2, 3,...}. 2) 2 The Coefficient Inequality for the class M α, λ, Ω) Theorem 1. Let 0 < α < 1, 0 λ < 1, Ω N 0 = {0, 1, 2, 3,...} and N = {1, 2, 3,...}. If fz) A satisfies the following coefficient inequality: ) k + Ω { ) )} k + k + 2α + λ) + + λ) )) k + a k+

3 Distortion bounds for a new subclass of analytic functions 3 + λ) 2α + λ) = { then fz) M α, λ, Ω). 2α ; 0 < α +λ λ α) ; +λ 2 α < + λ 3) Proof. In view of condition 2), we should show that ) 2α 1 λ)d Ω fz)) + λd Ω+1 fz)) + λ 1 λ) z DΩ fz))) + λ z 1 < 1. 4) DΩ+1 fz))) where Then We can see that ) 2α 1 λ)d Ω fz)) + λd Ω+1 fz)) + λ 1 λ) z DΩ fz))) + λ z 1 DΩ+1 fz))) = + λ) z + + λ) k+ A + B ) Ω+1 [ )] k+ A := [2α + λ)] z ) k + Ω [ )] [ k + B := 2α + λ) T 2α + λ) + k+ + λ) + λ) ) Ω ) [ 2α + λ) k+ k+ ) Ω+1 [ k+ 2α + λ) + < + λ) + λ) =: T, a k+ z k+ k + )] k+ a k+ z k ) Ω ) [ 2α + λ) k+ k+ With coefficient inequality 3), we can obtain ] a k+ ) k + Ω ) [ k + 2α + λ) k + )] a k+ z k+. )] k+ a k+ z k )] k+ a k+ )] a k+ 5)

4 4 Ibrahim Aktaş and Halit Orhan + λ) 2α + λ) where ) + k Ω+1 [ + λ) + k By using the inequality 6) in 5), we get ) 2α 1 λ)d Ω fz)) + λd Ω+1 fz)) + λ 1 λ) z DΩ fz))) + λ z 1 DΩ+1 fz))) < + λ) + λ) k+ C D ) Ω+1 [ )] = 1, k+ a k+ )] a +k. 6) or C := 2α + λ) + + λ) 2α + λ) ) k + Ω+1 [ )] k + D := + λ) a k+, 1 λ)d Ω fz)) + λd Ω+1 fz)) 1 λ) z DΩ fz))) + λ z + λ DΩ+1 fz))) 2α < + λ 2α that is fz) M α, λ, Ω). Theorem 2. If fz) M α, λ, Ω), then Re { 1 λ) z DΩ fz))) + λ z } DΩ+1 fz))) 1 λ)d Ω fz)) + λd Ω+1 > α fz)) + λ. z Proof. Let fz) M α, λ, Ω), Hz) = F z) F z) and F z) = 1 λ)d Ω fz)) + λd Ω+1 fz)). In this case, we can write that 1 Hz) + λ 2α < + λ 2α. After some calculations, we get 1 Hz) + λ 2α < + λ 2α = 1 Hz) + λ 2α 2α + λ) Hz) 2αHz) λ < 2α ) + λ 2 < = 2α + λ) Hz)) 2 < + λ) 2 Hz) 2 2α = 2α + λ) Hz)) 2α + λ) Hz)) < + λ) 2 Hz)Hz)) ) 2

5 Distortion bounds for a new subclass of analytic functions 5 = 4α 2 2α + λ) Hz) 2α + λ) Hz) + + λ) 2 Hz)Hz)) < + λ) 2 Hz)Hz)) We know that for each z C, { } = 4α 2 < 2α + λ) Hz) + Hz). 7) z + z = 2Rez. 8) If we consider equality 8) in inequality 7), then we obtain 2α < + λ) 2ReHz) = ReHz) > α + λ. This is desired. Remark 1. Let = 1. If fz) M 1 α, λ, Ω), then F z) S α 1+λ ). 3 Some Results Associated with Distortion Bounds In recent years, S. Owa, K. Ochiai and H. M. Srivastava [4] have introduced the integro-differential oerator for an analytic function fz) which is denoted in the form of I s fz) and defined as shown below: and I 1 fz) = f z), I 0 fz) = fz) I s fz) = z 0 I s 1 ft)dt for s N = {1, 2, 3,...}. Let us denote by M α, λ, Ω) the subclass of the class M α, λ, Ω) which satisfies the coefficient inequality 3) for some α and which consists of the fz) M α, λ, Ω). By definition in 1), we can write I s fz) =! s + )! zs+ + + k)! s + + k)! a +kz s++k 9) Now, by integro-differential oerator, we can obtain the following results about the distortion bounds for the functions belonging to the subclass M α, λ, Ω). Theorem 3. If fz) M α, λ, Ω), then we have the following inequality:! s + )! z +s s + + 1)! I s fz) )! { + λ) 2α + λ) } ) Ω ) { + λ) +1 ) + + λ) +1 )} z +s+1

6 6 Ibrahim Aktaş and Halit Orhan! s + )! z +s + s + + 1)! )! { + λ) 2α + λ) } ) Ω ) { + λ) +1 ) + + λ) +1 for z, s N { 1, 0}, 0 < α < 1, 0 λ < 1 and Ω N 0 = N {0}. )} z +s+1 Proof. After taking the absolute value in both sides of equality 9) and alying the triangle inequality, we can denote that I s fz) = < Besides, we can write! + k)! s + )! zs+ + s + + k)! a k+z s++k! s + )! z s+ + z s++1 + k)! s + + k)! a k+ z k 1! s + )! z s+ + z s++1 + k)! s + + k)! a k+. 10) ) Ω { ) )} s + + 1)! 2α + λ) + + λ) + 1)! 1 + λ ) + k)! s + + k)! a k+ ) + 1 Ω { ) )} α + λ) + + λ) 1 + λ ) a k+ ) + 1 Ω { ) )} α + λ) + + λ) )) k + a k+ + λ) 2α + λ) or + k)! s + + k)! a k+ s + + 1)! )! { + λ) 2α + λ) } ) Ω ) { + λ) +1 ) + + λ) +1 )}.

7 Distortion bounds for a new subclass of analytic functions 7 Using the last ste of inequality in 10), we obtain I s fz) + s + + 1)! +1! s + )! z s+ + 1)! { + λ) 2α + λ) } ) Ω ) { + λ) +1 As similar imlementations above, we can write I s fz) s + + 1)!! s + )! z s+ +1 ) + + λ) )! { + λ) 2α + λ) } z s++1 ) Ω ) { + λ) +1 By joining 11) and 12), we obtain! s + )! z s+ s + + 1)! +1 I s fz)! s + )! z s+ + s + + 1)! ) + + λ) )! { + λ) 2α + λ) } z s++1 ) Ω ) { + λ) )! { + λ) 2α + λ) } z s++1 ) Ω ) { + λ) )} z s++1 ) + + λ) +1 ) + + λ) +1 11) )} 12) )} )} Remark 2. If we choose λ = Ω = 0 and = 1 in Theorem 3, then we have the same results given by Owa, Ochiai and Srivastava [4]. Setting s = 1, 0, 1 in Theorem 3, we get the following Corollary.1. Corollary 1. If fz) M α, λ, Ω), then the following inequalities are obtained. z 1 ) Ω λ + 1) { + λ) 2α + λ) } ) { 2α + λ) +1 ) + + λ) +1 )} z f z) z 1 + ) Ω λ + 1) { + λ) 2α + λ) } ) { 2α + λ) +1 ) + + λ) +1 )} z

8 8 Ibrahim Aktaş and Halit Orhan for s = 1, z ) Ω λ z + for s = 0 and ) Ω λ { + λ) 2α + λ) } ) { 2α + λ) +1 ) + + λ) +1 { + λ) 2α + λ) } ) { 2α + λ) +1 ) + + λ) +1 )} z +1 fz) )} z +1 z { + λ) 2α + λ) } z +2 ) Ω ) { ) )} + 2) +1 + λ) λ) +1 I 1 fz) z { + λ) 2α + λ) } z +2 ) Ω ) { ) )}, + 2) +1 + λ) λ) +1 for s = 1. Remark 3. Putting = 1 in Corollary 1, we get the same result given by Kamali [7]. 4 Partial Sums for the Class M α, λ, Ω) In this section following the earlier works by Silverman [8], Deniz and Orhan [9], N. C. Cho. at al. [10] and others see also [11], [12]) on artial sums of analytic functions, we study the ratio of real arts of functions involving 1) and their sequence of artial sums defined by f n z) = z ; n = k ) n f n z) = z + a k+ z k+ ; n = k +, k + + 1,... and determine shar lower bounds for { } { } fz) fn z) Re, Re. f n z) fz) Theorem 4. Let fz) M α, λ, Ω) and f n z) be given by 1) and 13), resectively. Suose also that φ k, λ, α, Ω) a k+

9 Distortion bounds for a new subclass of analytic functions 9 where φ k = φ k, λ, α, Ω) ) k + Ω { ) )} k + k + = 2α + λ) + + λ) [ )] k + and =, λ, α) = + λ) 2α + λ). Then, we have { } fz) Re φ n+1 14) f n z) φ n+1 and Re { } fn z) φ n+1 fz) φ n ) This results are shar for every n with the extremal functions given by fz) = z + Proof. In order to rove 14), it suffices to show that { ϕ n+1 fz) f n z) ϕ } n z ϕ n+1 1 z We can write ϕ n+1 Then { fz) f n z) ϕ } n+1 ϕ n+1 Obviously w0) = 0 and = ϕ n+1 = wz) = n wz) 2 2 n 1 + ϕ n+1 ϕ n+1 φ n+1 z n ) z + a k+ z k+ z + n a k+ z k + ϕ n+1 ϕ n+1 z ). 17) ϕ n+1 a k+ z k+ ϕ n+1 a k+ z k + n a k+ z k 1 + n a k+ z k a k+ ϕ n+1 a k+ z k a k+ a k+ z k.. a k+ = 1 + wz) 1 wz).

10 10 Ibrahim Aktaş and Halit Orhan Now wz) 1 if and only if ϕ n+1 a k+ + n a k+ 1 18) It is suffices to show that the left hand side of 18) is bounded above by which is equivalent to n ϕ k ) a k+ + ϕ k ϕ n+1 ) a k+ 0. ϕ k a k+ To see that the function f given by 16) gives the shar result, we obseve for z = z e πi n+1 that fz) f n z) = 1 + ϕ n+1 z n+1 = 1 + We thus comlete the roof of inequality 14). ) z e πi n+1 φ n+1 n+1. ϕ n+1 φ n+1 The roof of inequality 15) can be made similar to that of 14), here we choose to omit the analogous details. References [1] Altıntaş, O., On a subclass of certain starlike functions with negative coefficients, Math. Jaonica ), no. 3, 1-7. [2] Duren, P. L., Univalent functions, Grundlehren der Mathematischen Wissenschaften 259, CitySringer-Verlag, StateNew York, StateBerlin, CityHeidelberg and CitylceTokyo, [3] Goodman, A. W., Univalent functions, Vol. I, Polygonal Publishing House, Statelace, Washington, Statelace New Jersey, [4] Owa, S., Ochiai, K., and Srivastava, H. M., Some Coefficient inequalities and distortion bounds associated with certain new subclasses of analytic functions, Mathematical Inequalities & Alications ), no.1, [5] Sălagean, G. S., Subclasses of univalent functions, Lecture Notes in Math. Sringer-Verlag ), [6] Silverman, H., Univalent functions with negative coefficients, Proc. Amer. Math. Soc ), [7] Kamali, M., Some results associated with distortion bounds and coefficient inequalities for certain new subclasses of analytic functions, Tbilisi Mathematical Journal ),

11 Distortion bounds for a new subclass of analytic functions 11 [8] Silverman, H., Patial sums of starlike and convex functions, Journal of Mathematical Analysis and Alications ), [9] Deniz, E., and Orhan, H., Some roerties of certain subclasses of analytic functions with negative coefficients by using generalized Ruscheweyh derivative oerator, Czechoslovak Mathematical Journal ), no.135, [10] Cho, N. C., and Owa, S., Partial sums of meromorhic functions, J. Ineq. Pure Al. Math Art. 30). Electronic only. [11] Raina, R. K., and Bansal, D., Some roerties of a new class of analytic functions defined in terms of a hadamard roduct., J. Ineq. Pure Al. Math Art. 22). Electronic only. [12] Latha, S., and Shivarudraa, L., Partial sums of meromorhic functions, J. Ineq. Pure Al. Math., Art. 140). Electronic only.

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