Research Article A Study on Becker s Univalence Criteria

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1 Abstract and Applied Analysis Volume 20, Article ID 75975, 3 pages doi:0.55/20/75975 Research Article A Study on Becker s Univalence Criteria Maslina Darus and Imran Faisal School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor D. Ehsan 43600, Malaysia Correspondence should be addressed to Maslina Darus, maslina@ukm.my Received 26 January 20; Accepted May 20 Academic Editor: Allan C. Peterson Copyright q 20 M. Darus and I. Faisal. 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. We study univalence properties for certain subclasses of univalent functions K, K 2, K 2,μ,andS p, respectively. These subclasses are associated with a generalized integral operator. The extended Becker-typed univalence criteria will be studied for these subclasses.. Introduction and Preliminaries Let A denote the class of analytic functions f in the open unit disk U {z : z < } normalized by f 0 f 0 0. Thus, each f A has a Taylor series representation f z z a k z k k 2. Let A 2 be the subclass of A consisting of functions of the form f z z a k z k k 3.2 Let K be the univalent subclass of A which satisfies z 2 f z ) 2 <, f z z U..3

2 2 Abstract and Applied Analysis Let K 2 be the subclass of K for which f 0 0. Let K 2,μ be the subclass of K 2 consisting of functions of the form.2 which satisfy z 2 f z ) 2 <μ, 0 <μ, z U..4 f z Next, we define a subclass S p of A consisting of all functions f z that satisfy z )) p, 0 <p 2, p R, z U..5 f z For functions f z z k 2 a kz k and g z z k 2 b kz k, the Hadamard product or convolution f g is defined as usual by ) f g z z a k b k z k k 2.6 Define the function ϕ a, c; z by a ϕ a, c; z z 2 F,a,c; z k z k, c k c/ 0,, 2,...,.7 k 0 where a k k is the famous Pochhammer symbol defined in terms of Gamma function. It is easily seen that ϕ 2 α, 2; z is a convex function, since zϕ z ϕ 2 α, ; z SV α Using the fractional derivative of order α, D α z, Owa and Srivastava 2 introduced the operator Ω α : A A which is known as an extension of fractional derivative and fractional integral, as follows: Ω α f z Γ 2 α z α Dzf z, α α/ 2, 3, 4,... Γ k Γ 2 α z a k z k Γ k α k 2 ϕ 2, 2 α; z f z.8 Note that Ω 0 f z f z For a function f in A, we define D n,ν α, β, μ f z : A A, the linear fractional differential operator, as follows: I,ν α, β, μ ) f z ν μ β I 0,ν ) α, β, μ f z f z, ) Ω α f z ) ) μ z Ω α f z ),

3 Abstract and Applied Analysis 3 ) ) ) α, β, μ f z I α α, β, μ f z I 2,ν I n,ν α, β, μ ) f z I α. I,ν I n,ν α, β, μ ) f z ).9 If f is given by., then by.8 and, weseethat I n,ν ) Γ k ) Γ 2 α ν μ ) )) n k β α, β, μ f z z a k z k.0 Γ k α k 2 From.8 and, D n,ν α, β, μ f z can be written in terms of convolution as I n,ν ) ] α, β, μ f z [ϕ 2, 2 α; z g μ,ν z ϕ 2, 2 α; z gμ,ν β, β, z f z, }{{}. where g μ, ν β, z z ν μ β ) / )) z 2 z 2 ) ν μ β z z 2 ) 2z 3z 2 z k 2 μ ) z 2 2 μ ) z 3. ) )) g μ, ν ν μ k β β, z z z k, ϕ 2, 2 α; z g μ,ν β, z ϕ 2, 2 α; z gμ,ν β, z n-times product,.3.2 which generalizes many operators. Indeed, if we choose suitably values of α, β, μ, and ν in.2, wehavethefollowing. i β, μ 0, and α 0, we obtain D m f z given by Aouf et al. 3. α, ii ν, β 0, μ 0, and α 0, we obtain D m f z given by Al-Oboudi 4. iii ν, β 0, μ 0,, and α 0, we obtain D m f z given by Sălăgean 5. iv ν, β,, μ 0, and α 0, we obtain I m f z given by Uralegaddi and Somanatha 6. v β,, μ 0, and α 0, we obtain I m l f z given by Cho and Srivastava 7 and Cho and Kim 8.

4 4 Abstract and Applied Analysis vi ν, β 0, μ 0, 0, and n, we obtain Owa and Srivastava differential operator 2. vii ν, β 0, and μ 0, we obtain D n,α f z given by Al-Oboudi and Al-Amoudi 9, 0. viii β l, μ 0, and α p,weobtaini n p, l f z given by Catas. ix β l, μ 0, α p, and, we obtain I n p, l f z given by Kumar et al. and Srivastava et al., respectively 2, 3. Next, we introduce a new family of integral operator by using generalized differential operator already defined above. For m N {0} and γ,γ 2,γ 3,...,γ n, ρ C\{0,, 2,...}, we define a family of integral operators Υ γi,η, n, ρ, ν, α, β : A m A m by Υ γi,η, ) z m I n,ν ) ) /γi n, ρ, ν, α, β : z ρ t ρ α, β, μ, η fi t dt t 0 /ρ, f i A,.4 which generalize many integral operators. In fact, if we choose suitable values of parameters in this type of operator, we get the following interesting operators. i ν, β 0, μ 0, α 0, γ i /α i,andρ, we obtain I f,...,f m given by Bulut 4. ii n 0, ν, β 0, μ 0, α 0, γ i / α,andρ n α, we obtain F n,α z given by Breaz et al. 5. iii n 0, ν, β 0, μ 0, α 0, γ i /α i,andρ, we obtain F α z given by D. Breaz and N. Breaz 6. For our main result, we need the following lemmas. Lemma. see 7, 8. Let c be a complex number, c, c /.Iff z z a 2 z 2 is a regular function in U and c z 2 z 2) zf z f z, z U,.5 then the function f is regular and univalent in U. Lemma.2 Schwarz Lemma. Let the function f z be regular in the disk U R {z C : z <R} with f z <M.Iff z has one zero with multiply m for z 0, then f z M R m z m, z U R,.6 and equality holds only if f z e iθ M/R m z m,whereθ is constant.

5 Abstract and Applied Analysis 5 Lemma.3 see 9. Let δ be a complex number with Re δ>0 such that c C, c, c/. If f A satisfies the condition then the function is analytic and univalent in U. c z 2δ z 2δ) zf z δf z, z U,.7 F δ z { δ z 0 t δ f t dt} /δ.8 Lemma.4 see 20. If a function f S p,then z 2 f z ) 2 f z p z 2, z U Univalence Properties In this section, we will discuss the univalence properties of the new family of integral operators mentioned above. Theorem 2.. Let c be a complex number, I n,ν α, β, μ, η f i z M i, M i for all i {, 2, 3,...} and I n,ν α, β, μ, η f i t S p i for i {, 2, 3,...} such that R ρ ) Mi p i 2 ) M i, 2. γi Mi then the family Υ γi,η, n, ρ, ν, α, β : z is univalent. Re ρ ) Mi p i 2 ) M i, M i, 2.2 γi Mi Proof. Since I n,ν α, β, μ, η f i t S p i,sobylemma.4, we have z 2 I n,ν ) α, β, μ, η fi t ) n,ν ) α, β, μ, η fi t ) 2 p i z 2, z U. 2.3 Now, by using hypothesis, we have n,ν ) α, β, μ, η fi z Mi, 2.4

6 6 Abstract and Applied Analysis so by Lemma.3, weget n,ν ) α, β, μ, η fi z Mi z, R. 2.5 Let I n,ν ) α, β, μ f z Γ k ) Γ 2 α ν μ ) )) n k β a k z k / 0 z Γ k α k 2 if z 0, 2.6 so m I n,ν ) ) /γi α, β, μ, η fi z I n,ν z α, β, μ, η ) f z z ) /γ I n,ν α, β, μ, η ) fm z z ) /γm. 2.7 Let F z I n,ν ) ) /γ α, β, μ, η f t I n,ν ) ) /γm α, β, μ, η fm t dt, t t z which implies that I n,ν ) ) /γ F z α, β, μ, η f t I n,ν ) ) /γm α, β, μ, η fm t t t zf z F z γ z I n,ν ) α, β, μ, η f z ) ) ) α, β, μ, η f z I n,ν γ m z I n,ν ) α, β, μ, η fm z ) ) ), α, β, μ, η fm z I n,ν 2.9 zf z F z m z I n,ν ) α, β, μ, η fi z ) ) ). 2.0 α, β, μ, η fi z γ i I n,ν This implies that F z z I n,ν ) α, β, μ, η fi z ) ) ) γ i α, β, μ, η fi z, 2. I n,ν

7 Abstract and Applied Analysis 7 or F z z I n,ν ) α, β, μ, η fi z ) γ i n,ν ) α, β, μ, η fi z ) I n,ν ) α, β, μ, η fi z ) 2 z 2.2 Using 2.5, weget F z z I n,ν ) α, β, μ, η fi z ) γ i n,ν ) α, β, μ, η fi z ) 2 M i 2.3 This implies that F z z I n,ν ) α, β, μ, η fi z ) γi n,ν ) α, β, μ, η fi z ) 2 M i M i 2.4 By using 2.3, weget F z ) p γi i z 2 M i M i, 2.5 which implies that F z γ i p i M i M i M 2 i M3 i ) ), 2.6 because M i,m 2 i,m3 i,..., implies that ) ) F z Mi )) 2Mi p γi i M i p M i γi i M i, M i F z pi M 2 i p )) im i 2M i pi M i p i 2 ) )) M i, γ i M i γ i M i F z Mi p i 2 ) )) M i γi M i 2.7 Now, we calculate c z 2ρ z 2ρ) zf z ρf z c zf z ρ F z c R ρ ) F z. 2.8

8 8 Abstract and Applied Analysis This implies that c z 2ρ z 2ρ) zf z ρf z < c R ρ ) Mi p i 2 ) )) M i 2.9 γi M i By using 2.46, we conclude that c z 2ρ z 2ρ) zf z ρf z c zf z ρ F z 2.20 Hence, by Lemma.3, the family of integral operators Υ γi,η, n, ρ, ν, α, β : z is univalent. Corollary 2.2. Let c be a complex number, I n,ν α, β, μ, η f i z M, M for all i {, 2, 3,...} and I n,ν α, β, μ, η f i t S p, M i M, for all i {, 2, 3,...} such that R ρ ) M p 2 ) M γi M, 2.2 M Re ρ ) pi 2 ) M, M, 2.22 γi M then the family Υ γi,η, n, ρ, ν, α, β : z is univalent. Corollary 2.3. Let c be a complex number, I n,ν α, β, μ, η f i z M, M, for all i {, 2, 3,...} and the family I n,ν α, β, μ, η f i t S p, M i M, γ i γ, for all i {, 2, 3,...} such that R ρ ) M p 2 ) M γ M, 2.23 M Re ρ ) pi 2 ) M γ, M, 2.24 M then the family Υ γi,η, n, ρ, ν, α, β : z is univalent. Using the method given in the proof of Theorem 2., one can prove the following results.

9 Abstract and Applied Analysis 9 Theorem 2.4. Let c be a complex number, I n,ν α, β, μ, η f i z M i, M i for all i {, 2, 3,...} and the family I n,ν α, β, μ, η f i t S p i for i {, 2, 3,...} and c such that R ρ ) pi M i ) M i γi pi M i, 2.25 then the family Υ γi,η, n, ρ, ν, α, β : z is univalent. Re ρ ) pi M i ) M i ), M γ i i, 2.26 pi M i Theorem 2.5. Let c be a complex number, I n,ν α, β, μ, η f i z M i, M i for all i {, 2, 3,...,n} and I n,ν α, β, μ, η f i t S p i for i {, 2, 3,...,n} such that R ρ ) pi M i M n i 2 ) M i, 2.27 γi Mi then the family Υ γi,η, n, ρ, ν, α, β : z is univalent. Re ρ ) pi M i M n i 2 ) M i, M i, 2.28 γ Mi i Theorem 2.6. Let c be a complex number, I n,ν α, β, μ, η f i z M i, M i for all i {, 2, 3,...,n} and I n,ν α, β, μ, η f i t S p i for i {, 2, 3,...,n} such that R ρ ) pi n n /2 ) M i γi, 2.29 then the family Υ γi,η, n, ρ, ν, α, β : z is univalent. Re ρ ) pi n n /2 ) M i, M γi i, 2.30 Theorem 2.7. Let c be a complex number, I n,ν α, β, μ, η f i z M i, M i for all i {, 2, 3,...,n} and I n,ν α, β, μ, η f i t K 2,μi,fori {, 2, 3,...,n} such that R ρ ) μi n n ) M i γ i, 2.3

10 0 Abstract and Applied Analysis Re ρ ) μi n n ) M i, M γi i, 2.32 then the family Υ γi,η, n, ρ, ν, α, β : z is univalent. Proof. Using the proof of Theorem 2., we have F z z I n,ν ) α, β, μ, η fi z ) γi n,ν ) α, β, μ, η fi z ) 2 M i Since I n,ν α, β, μ, η f i t K 2,μi,sobyusing.4, weget z 2 I n,ν ) α, β, μ, η fi z ) n,ν ) α, β, μ, η fi z ) 2 <μ i, 0 <μ, z U So from 2.33, weget or F z z I n,ν ) α, β, μ, η fi z ) γi n,ν ) α, β, μ, η fi z ) 2 M i M i, 2.35 F z ) μi M γi i 2M i, Mi >, F z μi M γi i 2M i 4M i n-times ), M i >, F z ) μi M γi i n n M i, Mi > Now, we evaluate the expression c z 2ρ z 2ρ) zf z zf ρf z c z ρ F z c R ρ ) F z, c z 2ρ z 2ρ) zf z ρf z c R ρ ) ) μi M γ i i n n M i. 2.37

11 Abstract and Applied Analysis Using 2.45 and 2.46, we conclude that c z 2ρ z 2ρ) zf z ρf z Hence by using Lemma.3, the family Υ γi,η, n, ρ, ν, α, β : z is univalent. Corollary 2.8. Let c be a complex number, I n,ν α, β, μ, η f i z M, M for all i {, 2, 3,...,n} and I n,ν α, β, μ, η f i t K 2,μi,fori {, 2, 3,...,n} such that R ρ ) μi n n ) M γi, 2.39 Re ρ ) μi n n ) M, M, 2.40 γi then the family Υ γi,η, n, ρ, ν, α, β : z is univalent. Corollary 2.9. Let c be a complex number, I n,ν α, β, μ, η f i z M, M, γ i γ for all i {, 2, 3,...,n} and I n,ν α, β, μ, η f i t K 2,μi,fori {, 2, 3,...,n} such that R ρ ) μi n n ) M γ, 2.4 Re ρ ) μi n n ) M γ, M, 2.42 then the family Υ γi,η, n, ρ, ν, α, β : z is univalent. Using a similar method as in the proof of Theorem 2.7, one can prove the following results. Theorem 2.0. Let c be a complex number, I n,ν α, β, μ, η f i z M i, M i for all i {, 2, 3,...,n} and I n,ν α, β, μ, η f i t K 2,μi,fori {, 2, 3,...,n} such that R ρ ) μi M i ) M i M n i M i, 2.43 γ Mi i

12 2 Abstract and Applied Analysis Re ρ ) μi M i ) M i M n i M i, M i, 2.44 γi Mi then the family Υ γi,η, n, ρ, ν, α, β : z is univalent. Theorem 2.. Let c be a complex number, I n,ν α, β, μ, η f i z M i, M i for all i {, 2, 3,...} and I n,ν α, β, μ, η f i t K 2,μi,fori {, 2, 3,...} such that R ρ ) μi M i μ i 2 ) M i, 2.45 γ Mi i Re ρ ) μi M i μ i 2 ) M i, M i, 2.46 γi Mi then the family Υ γi,η, n, ρ, ν, α, β : z is univalent. Note that some other related work involving integral operators regarding univalence criteria can also be found in Acknowledgment The work presented here was partially supported by UKM-ST-06-FRGS References S. Owa, On the distortion theorems. I, Kyungpook Mathematical Journal, vol. 8, no., pp , S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canadian Mathematics, vol. 39, no. 5, pp , M. K. Aouf, R. M. El-Ashwah, and S. M. El-Deeb, Some inequalities for certain p-valent functions involving extended multiplier transformations, Proceedings of the Pakistan Academy of Sciences, vol. 46, no. 4, pp , F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, International Mathematics and Mathematical Sciences, no , pp , G. Ş. Sălăgean, Subclasses of univalent functions, in Complex Analysis 5th Romanian-Finnish Seminar, Part Bucharest, 98), vol. 03 of LectureNotesin Mathematics, pp , Springer, Berlin, Germany, B. A. Uralegaddi and C. Somanatha, Certain classes of univalent functions, in Current Topics in Analytic Function Theory, pp , World Scientific, Singapore, N. E. Cho and H. M. Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Mathematical and Computer Modelling, vol. 37, no. -2, pp , N. E. Cho and T. H. Kim, Multiplier transformations and strongly close-to-convex functions, Bulletin of the Korean Mathematical Society, vol. 40, no. 3, pp , F. M. Al-Oboudi and K. A. Al-Amoudi, On classes of analytic functions related to conic domains, Mathematical Analysis and Applications, vol. 339, no., pp , 2008.

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