SOME RESULTS OF p VALENT FUNCTIONS DEFINED BY INTEGRAL OPERATORS. Gulsah Saltik Ayhanoz and Ekrem Kadioglu

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1 Acta Universitatis Aulensis ISSN: No. 32/ SOME ESULTS OF VALENT FUNCTIONS EFINE BY INTEAL OPEATOS ulsah Saltik Ayhanoz and Ekre Kadioglu Abstract. In this aer, we derive soe roerties for,n,l,δ z) and F,n,l,δ z) considering the classes MT, β i, µ i ), K, β i, µ i ) and N γ). Two new subclasses KF,n,l β, µ, δ 1, δ 2,..., δ n ) and K,n,l β, µ, δ 1, δ 2,..., δ n ) are defined. Necessary and sufficient conditions for a faily of functions f i and g i, resectively, to be in the KF,n,l β, µ, δ 1, δ 2,..., δ n ) and K,n,l β, µ, δ 1, δ 2,..., δ n ) are defined. As secial z n z n ) δ cases, the roerties of f t)) δ dt and dt are given. 0 0 ft) t 2000 Matheatics Subject Classification: 30C45. Key Words and Phrases. Analytic functions; Integral oerators; β uniforly valently starke and β uniforly valently convex functions. Let A denote the class of the for 1. Introduction and reinaries fz) = z + a z, N = 1, 2,..., ), 1) which are analytic in the oen disc U = z C : z < 1. Also denote T the subclass of A consisting of functions whose nonzero coefficients, fro the second one, are negative and has the for fz) = z a z, a 0, N = 1, 2,..., ). 2) Also A 1 = A, T 1 = T. A function f A is said to be valently starke of order α 0 α < ) if and only if zf z) > α, z U). fz) 69

2 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... We denote by Sα), the class of all such functions. On the other hand, a function f A is said to be valently convex of order α 0 α < ) if and only if 1 + zf z) f > α, z U). z) Let C α) denote the class of all those functions which are valently convex of order α in U. Note that S0) = S and C 0) = C are, resectively, the classes of valently starke and valently convex functions in U. Also, we note that S1 0) = S and C 1 = C are, resectively, the usual classes of starke and convex functions in U. Let N γ) be the subclass of A consisting of the functions f which satisfy the inequaty 1 + zf z) f < γ, z U), γ >. 3) z) Also N 1 γ) = N γ). For = 1, this class was studied by Owa see [12]) and Mohaed see [9]). For a function f A, we define the following oerator 0 fz) = fz) 1 fz) = 1 zf z). ) k fz) = k 1 fz), 4) where k N 0 = N 0. The differential oerator k was introduced by Shenan et al. see [18]). When = 1, we get Sălăgean differential oerator see [15]). We note that if f A, then k fz) = z + We also note that if f T, then k fz) = z ) k a z, N = 1, 2,...) z U). ) k a z, N = 1, 2,...) z U). 70

3 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... Let MT, β, µ) be the subclass of A consisting of the functions f which satisfy the analytic characterization z fz) ) l i fz) < β µz fz) ) l + i fz), 5) for soe 0 < β, 0 µ <, N 0 = N 0 and z U. For = 1, l 1 = l 2 =... = l n = 0 for al = 1, 2,..., n this class was studied see [2]). efinition 1. A function f A is said to be in the class K, β, µ) if satisfies the following inequaty: 1 + z fz) ) l µ i fz)) 1 + z fz) ) l + β, 6) i fz)) for soe 0 β <, µ 0, N 0 = N 0 and z U. For = 1, l 1 = l 2 =... = l n = 0 for al = 1, 2,..., n this class was studied see [17], [9]). efinition 2. Let l = l 1, l 2,..., l n ) N n 0, δ = δ 1, δ 2,..., δ n ) n + for all i = 1, 2,..., n, n N. We define the following generantegral oerators I l,δ,n f 1, f 2,..., f n ) : A n A and I l,δ,n f 1, f 2,..., f n ) = F,n,l,δ z), z n F,n,l,δ z) = t 1 ) δi f i t) dt, 7) 0 t J l,δ,n g 1, g 2,..., g n ) : A n A J l,δ,n g 1, g 2,..., g n ) =,n,l,δ z),,n,l,δ z) = z n t 1 0 ) g i t) ) δi dt, 8) t 1 where f i, g i A for al = 1, 2,..., n and is defined by 4). eark 1. 7) integral oerator was studied and introduced by Saltık et al. see [16]). We note that if l 1 = l 2 =... = l n = 0 for al = 1, 2,..., n, then the integral oerator F,n,l,δ z) reduces to the oerator F z) which was studied by Frasin see [6]). Uon setting = 1 in the oerator 7), we can obtain the integral oerator k F z) which was studied by Breaz see [5]) and Breaz see [4]). For = 1 and l 1 = l 2 =... = l n = 0 in 7), the integral oerator F,n,l,δ z) reduces to the oerator F n z) which was studied by Breaz, Breaz see [2]) and Mohaed see 71

4 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... [10]). Observe that = n = 1, l 1 = 0 and δ 1 = δ, we obtain the integral oerator I δ f)z) which was studied by Pescar and Owa see [13]),. Breaz see [5]) and Mohaed see [11]) for δ 1 = δ [0, 1] secial case of the oerator I δ f)z) was studied by Miller, Mocanu and eade see [8]). For = n = 1, l 1 = 0 and δ 1 = 1 in 7), we have Alexander integral oerator If)z) in see [1]). eark 2. 8) integral oerator was studied and introduced by Saltık et al. see [16]). For l 1 = l 2 =... = l n = 0 in 8) the integral oerator,n,l,δ z) reduces to the oerator z) which was studied by Frasin see [6]). For = 1 and l 1 = l 2 =... = l n = 0 in 8), the integral oerator,n,l,δ z) reduces to the oerator δ1,δ 2,...,δ n z) which was studied by Breaz, Breaz and Owa see [3]) and Mohaed see [10]). Observe = n = 1, l 1 = 0 and δ 1 = δ, we obtain the integral oerator z) which was introduced and studied by Pfaltzgraff see [14]), Mohaed see [11]),.Breaz see [5]) and Ki and Merkes see [7]). Now, by using the equations 7) and 8) and the efinition 1 we introduce the following two new subclasses of K, β, µ). efinition 3. A faily of functions f i, i = 1, 2,..., n is said to be in the class KF,n,l β, µ, δ 1, δ 2,..., δ n ) if satisfies the inequaty: 1 + z F,n,l,δ z) ) F,n,l,δ z)) µ 1 + z F,n,l,δ z) ) F,n,l,δ z)) + β, 9) for soe 0 β <, µ 0, N 0 = N 0 and z U where F,n,l,s defined in 7). efinition 4. A faily of functions g i, i = 1, 2,..., n is said to be in the class K,n,l β, µ, δ 1, δ 2,..., δ n ) if satisfies the inequaty: 1 + z,n,l,δ z) ),n,l,δ z)) µ 1 + z,n,l,δ z) ),n,l,δ z)) + β, 10) for soe 0 β <, µ 0, N 0 = N 0 and z U where,n,l,s defined in 8). 2. Sufficient conditions of the oerator F,n,l,δ z) First, in this section we rove a sufficient condition for the integral oerator F,n,l,δ z) to be in the class N η). Theore 1. Let l = l 1, l 2,..., l n ) N n 0, δ = δ 1, δ 2,..., δ n ) n +, 0 µ i <, 0 < β i and f i A for al = 1, 2,..., n. If MT, β i, µ i ), then the integral oerator 72 f i) z) f i z) < M i and f i

5 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... is in N η), where F,n,l,δ z) = η = z 0 t 1 n ) δi f i t) dt, t β i + µ i M i ) +. 11) Proof. Fro the definition 7), we observe that F,n,l,δ z) A. On the other hand, it is easy to see that z, we obtain 1 + F,n,l,δ z) = z 1 n ) δi f i z). 12) z Now, we differentiate 12) logarithically and ultily by,n,l,δ z) ) z F,n,l,δ z) = + f i z) fi z) We calculate the real art fro both ters of the above exression and obtain 1 + Since w w, then 1 +,n,l,δ z) F,n,l,δ z),n,l,δ z) F,n,l,δ z) = ) ) z f i z) l +. i fi z) z f i ) z) fi z) Since f i MT, β i, µ i ) for al = 1, 2,..., n, we have 1 + Since 13) and,n,l,δ z) F,n,l,δ z) f i) z) f i z) β i µ i z f i ) z) fi z) β i µ i z f i ) z) fi z) < M i, we obtain , 13) + β i +. 73

6 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent ,n,l,δ z) F,n,l,δ z) < β i µ i M i + β i + = Hence F,n,l,δ z) N η), η = n β i + µ i M i ) +. β i + µ i M i ) +. eark 3.For = 1, = 0 for al = 1, 2,..., n in Theore 1, we obtain Theore 1 see [9]). Putting = 1 in Theore 1, we have Corollary 1. Let l = l 1, l 2,..., l n ) N n 0, δ = δ 1, δ 2,..., δ n ) n +, 0 µ i < 1, 0 < β i 1 and f i A for al = 1, 2,..., n. If MT 1, β i, µ i ) then the integral oerator is in N η), where F 1,n,l,δ z) = η = z 0 n ) δi f i t) dt, t β i 1 + µ i M) + 1. f i) z) f i z) < M i and f i Putting = n = 1, l 1 = 0, δ 1 = δ, µ 1 = µ, β 1 = β, M 1 = M and f 1 = f in Theore 1, we have Corollary 2. Let δ +, 0 µ < 1, 0 < β 1 and f A. If f z) fz) < M and f MT 1, β, µ) then the integral oerator ) z δ ft) 0 t is in N η), where η = δβ 1 + µm) Sufficient conditions of the oerator,n,l,δ z) Next, in this section we give a condition for the integral,n,l,δ z) to be valently convex. Theore 2. Let l = l 1, l 2,..., l n ) N n 0, δ = δ 1, δ 2,..., δ n ) n +, µ i 0, g i K, β i, µ i ) and let β i 0 be real nuber with the roerty 0 β i < for al = 1, 2,..., n. Moreover suose that 0 < n β i ), then the integral oerator,n,l,δ z) = z n t ) g i t) ) δi dt, t 1

7 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... is convex order of σ = n β i ). Proof. Fro the definition 8), we observe that,n,l,δ z) A. On the other hand, it is easy to see that,n,l,δ z) = z 1 n ) g i z) ) δi. 14) z 1 Now, we differentiate 14) logarithically and ake the siilar oerators to the roof of the Theore 2, we obtain 1 + z,n,l,δ z) z,n,l,δ z) = + g i z) ) ) gi z)) + 1. We calculate the real art fro both ters of the above exression and obtain 1 + z,n,l,δ z),n,l,δ z) = 1 + z g i z) ) gi z)) +. Since g i K, β i, µ i ) for al = 1, 2,..., n,we have 1 + z,n,l,δ z),n,l,δ z) > µ i 1 + z g i z) ) ) gi z)) + β i Since µ i 1 + zg i z)) g i z)) > 0, we obtain 1 + z,n,l,δ z),n,l,δ z) β i ), +. which ies that,n,l,δ z) is valently convex of order σ = n β i ). eark 4. Setting = 1, = 0 and g i = f i for al = 1, 2,..., n in Theore 2, we have obtain Theore 2 in see [9]). Putting = 1 in Theore 2, we have Corollary 3. Let l = l 1, l 2,..., l n ) N n 0, δ = δ 1, δ 2,..., δ n ) n +, µ i 0, g i K 1, β i, µ i ) and let β i 0 be real nuber with the roerty 0 β i < 1 for al = 1, 2,..., n. Moreover suose that 0 < n 1 β i ) 1, then the integral oerator 1,n,l,δ z) = z 0 n g i t) ) ) dt is convex order of σ = 1 n 1 β i ). 75

8 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... Putting = n = 1, l 1 = 0, δ 1 = δ, µ 1 = µ, β 1 = β and g 1 = g in Theore 2, we have Corollary 4. Let δ +, µ 0, g K 1, β, µ) and let β 0 be real nuber with the roerty 0 β < 1. Moreover suose that 0 < δ 1 β) 1, then the integral oerator 1,1,0,δ z) = z 0 g t)) δ dt is convex order of σ = 1 δ 1 β). 4. A necessary and suffficient condition for a faily of analytic functions f i KF,n,l β, µ, δ 1, δ 2,..., δ n ) In this section, we give a necessary and sufficient condition for a faily of functions f i KF,n,l β, µ, δ 1, δ 2,..., δ n ). Before ebarking on the roof of our result, let us calculate the exression ecall that, fro 7), we have F,n,l,δ,n,l,δ z) F,n,l,δ z) = z 1 n z), required for roving our result. ) δi f i z). 15) Now, we differentiate 15) logarithically and ultily by z, we obtain 1 + Let f i z) = z,n,l,δ z) F,n,l,δ z) = f i ) z) = z ) a,i z,,n,l,δ z) F,n,l,δ z) = = z z f i ) z) fi z) ) a,i z 1 and we get z z ). ) a,i z ) a,i z, 16) ) ) a,i z 1 ) a,i z. Theore 3. Let the function f i T for i 1, 2,..., n. Then the functions 76

9 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... f i KF,n,l β, µ, δ 1, δ 2,..., δ n ) for i 1, 2,..., n if and only if ) δ i ) µ + 1) a,i 1 ) β. 17) a,i Proof. First consider,n,l,δ µ 1 + z) F,n,l,δ z) 1 + Fro 16), we obtain,n,l,δ z) F,n,l,δ z),n,l,δ µ + 1) 1 + z) F,n,l,δ z).,n,l,δ µ + 1) 1 + z) F,n,l,δ z), ) ) a,i z = µ + 1) 1 ), a,i z ) δ i ) a,i z µ + 1) 1 ) a,i z, µ + 1) ) δ i ) a,i 1 ). a,i If 17) holds then the above exression is bounded by β and consequently,n,l,δ µ 1 + z) F,n,l,δ z),n,l,δ 1 + z) F,n,l,δ z) < β, which equivalent to 1 +,n,l,δ z) F,n,l,δ z),n,l,δ µ 1 + z) F,n,l,δ z) + β. 77

10 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... Hence f i KF,n,l β, µ, δ 1, δ 2,..., δ n ) for i 1, 2,..., n. Conversely, let f i KF,n,l β, µ, δ 1, δ 2,..., δ n ) for i 1, 2,..., n and rove that 17) holds. If f i KF,n,l β, µ, δ 1, δ 2,..., δ n ) for i 1, 2,..., n and z is real, we get fro 7) and 16) µ µ That is equivalent to β. ) ) a,i z 1 ) a,i z, ) ) a,i z 1 ) + β, a,i z ) ) a,i z 1 ) a,i z + β. ) µ ) a,i z 1 ) a,i z + ) δ i ) a,i z 1 ) a,i z, The above inequaty reduce to ) µ + 1) ) a,i z 1 ) a,i z β. Let z 1 along the real axis, then we get ) µ + 1) ) a,i 1 ) β, a,i 78

11 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... which give the required result. eark 5. Setting = 1, = 0 for i 1, 2,..., n in Theore 3, we have obtain Theore 3 in see [9]). Putting = 1 in Theore 3, we have Corollary 5.Let the function f i T for i 1, 2,..., n. Then the functions f i KF 1,n,l β, µ, δ 1, δ 2,..., δ n ) for i 1, 2,..., n if and only if ) 1) µ + 1) a,i =2 1 ) l 1 β. i a,i =2 Putting = n = 1, l 1 = 0, δ 1 = δ and f 1 = f in Theore 3, we have Corollary 6. Let the function f T. Then the functions f KF 1,1,0 β, µ, δ) if and only if =2 δ 1) µ + 1) a,1 1 =2 a,1 1 β. 5.A necessary and suffficient condition for a faily of analytic functions g i K,n,l β, µ, δ 1, δ 2,..., δ n ) In this section, we give a necessary and sufficient condition for a faily of functions g i K,n,l β, µ, δ 1, δ 2,..., δ n ). Let us calculate the exression z,n,l,δ z) required for roving our result. ecall that, fro 8), we have and,n,l,δ z) = z 1 n,n,l,δ z), ) g i z) ) δi. 18) z 1 Now, we differentiate 18) logarithically and ultily by z, we obtain 1 + z,n,l,δ z) z,n,l,δ z) = g i z) ) ) gi z)) + 1. Let g i z) = z ) a,i z, g i ) z) = z 1 ) a,i z 1 79

12 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... g i ) z) = 1) z 2 ) 1) a,i z 2, we = =,n,l,δ z) 1 + z,n,l,δ 19) z) 1) z 1 ) 1) a,i z 1 z 1 ) + 1 a,i z 1, n ) ) a,i z ) a,i z. Theore 4. Let the function g i T for i 1, 2,..., n. Then the functions g i K,n,l β, µ, δ 1, δ 2,..., δ n ) for i 1, 2,..., n if and only if ) δ i ) µ + 1) a,i ) β. 20) a,i Proof. First consider µ 1 + z,n,l,δ z),n,l,δ z) 1 + z,n,l,δ z),n,l,δ z) µ + 1) 1 + z,n,l,δ z),n,l,δ z). Fro 19), we obtain 80

13 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... µ + 1) 1 + z,n,l,δ z),n,l,δ z) ) ) a,i z = µ + 1) ), a,i z ) δ i ) a,i z µ + 1) ) a,i z, µ + 1) ) δ i ) a,i ). a,i If 20) holds then the above exression is bounded by β and consequently µ 1 + z,n,l,δ z),n,l,δ z) 1 + z,n,l,δ z),n,l,δ z) < β, which equivalent to 1 + z,n,l,δ z),n,l,δ z) µ 1 + z,n,l,δ z),n,l,δ z) + β. Hence g i K,n,l β, µ, δ 1, δ 2,..., δ n ) for i 1, 2,..., n. Conversely, let g i K,n,l β, µ, δ 1, δ 2,..., δ n ) for i 1, 2,..., n and rove that 20) holds. If g i K,n,l β, µ, δ 1, δ 2,..., δ n ) for i 1, 2,..., n and z is real, we get fro 8) and 19) 81

14 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... µ µ That is equivalent to β. ) ) a,i z ) a,i z, ) ) a,i z ) + β, a,i z ) ) a,i z ) a,i z + β. ) µ ) a,i z ) a,i z + ) δ i ) a,i z ) a,i z, The above inequaty reduce to ) µ + 1) ) a,i z ) a,i z β. Let z 1 along the real axis, then we get n ) )a,i µ+1) ) a,i β, which give the required result. eark 6. Setting = 1, = 0 for i 1, 2,..., n in Theore 4, we have obtain Theore 4 in see [9]). Putting = 1 in Theore 4, we have 82

15 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... Corollary 7. Let the function g i T for i 1, 2,..., n. Then the functions g i K 1,n,l β, µ, δ 1, δ 2,..., δ n ) for i 1, 2,..., n if and only if µ + 1) ) 1) a,i =2 1 ) l 1 β. i a,i =2 Putting = n = 1, l 1 = 0, δ 1 = δ and g 1 = g in Theore 4, we have Corollary 8. Let the function g T. Then the functions g K 1,1,0 β, µ, δ) if and only if eferences δµ+1) 1)a,1 =2 1 =2 a,1 1 β. [1] J. W. Alexander, Functions which a the interior of the unit circle uon sile regions, Annals of Matheatics, 17 1) 1915) [2]. Breaz and N. Breaz, Two integral oerators, Studia Universitatis Babeş- Bolyai. Matheatica, 47 3) 2002) [3]. Breaz, S. Owa and N. Breaz, A new integral univalent oerator, Acta Univ. Aulensis Math. Infor., ) [4]. Breaz, H. Ö. üney and. S. Sălăgean, A new generantegral oerator, Tasui Oxford Journal of Matheatical Sciences, 25 4) 2009) [5]. Breaz, Certain integral oerators on the classes Mβ i ) and Nβ i ), Journal of Inequaties and Acations, vol. 2008, Article I [6] B. A. Frasin, Convexity of integral oerators of valent functions, Math. Cout. Model., ) [7] Y. J. Ki and E. P. Merkes, On an integral of owers of a siralke function, Kyungook Matheatical Journal, vol ) [8] S. S. Miller, P. T. Mocanu, and M. O. eade, Starke integral oerators, Pacific Journal of Matheatics, 79 1) 1978) [9] A. Mohaed, M. arus and. Breaz, Soe roerties for certain integral oerators, Acta Universitatis Aulensis, ),

16 ulsah Saltik Ayhanoz and Ekre Kadioglu-Soe esults of -valent... [10] A. Mohaed, M. arus and. Breaz, On close to convex for certain integral oerators, Acta Universitatis Aulensis, No 19/2009, [11] A. Mohaed, M. arus and. Breaz, Fractional Calculus for Certain Integral oerator Involving Logarithic Coefficients, Journal of Matheatics and Statics, 5: ), [12] S. Owa and H.M. Srivastava, Soe generazed convolution roerties associated with certain subclasses of analytic functions, JIPAM, 3 3) 2003), 42: [13] V. Pescar and S. Owa, Sufficient conditions for univalence of certain integral oerators, Indian Journal of Matheatics, 42 3) 2000) [14] J. A. Pfaltzgraff, Univalence of the integral of f z)) λ, Bull. London Math. Soc. 7 3) 1975) [15]. S. Sălăgean, Subclasses of univalent functions, Lecture Notes in Matheatics Sringer Verlag, Bern Heidelberg and NewYork 1983) [16]. Saltık, E. eniz and E. Kadıoğlu, Two New eneral valent Integral Oerators, Math. Cout. Model., ) [17] S. Shas, S.. Kulkarni and J. M. Jahangiri, Classes of uniforly starke and convex functions, Internat. J. Math. Sci., ) [18]. M. Shenan, T. O. Sa and M. S. Marouf, A certain class of ultivalent restarke functions involving the Srivastava-Saigo-Owa fractionantegral oerator, Kyungook Math. J ) ülşah Saltık Ayhanöz eartent of Matheatics University of Atatürk Erzuru, eail:gsaltik@atauni.edu.tr Ekre Kadıoğlu eartent of Matheatics University of Atatürk Erzuru, eail:ekre@atauni.edu.tr 84

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