A Certain Subclass of Analytic Functions Defined by Means of Differential Subordination

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1 Filomat 3:4 6), DOI.98/FIL64743S Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: A Certain Subclass of Analytic Functions Defined by Means of Differential Subordination H. M. Srivastava a, Dorina Răducanu b, Paweł Zaprawa c a Department of Mathematics Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada China Medical University, Taichung 44, Taiwan, Republic of China b Faculty of Mathematics Computer Science, Transilvania University of Braşov, Iuliu Maniu 5, R-59 Braşov, Romania c Department of Mathematics, Lublin University of Technology, Nadbystrycka 38D, PL--68 Lublin, Pol Abstract. For α π, π], let R α φ) denote the class of all normalied analytic functions in the open unit disk U satisfying the following differential subordination: f ) + + e iα ) f ) φ) U), where the function φ) is analytic in the open unit disk U such that φ) =. In this paper, various integral convolution characteriations, coefficient estimates differential subordination results for functions belonging to the class R α φ) are investigated. The Fekete-Segö coefficient functional associated with the kth root transform [ f k )] /k of functions in R α φ) is obtained. A similar problem for a corresponding class R Σ;α φ) of bi-univalent functions is also considered. Connections with previous known results are pointed out.. Introduction Let A denote the class of functions f ) of the form: f ) = + a n n n= U). ) which are analytic in the open unit disk U = { : C < }. We denote by S the subclass of A consisting of univalent functions in U by C the familiar subclass of S whose members are convex functions in U. Mathematics Subject Classification. Primary 3C45; Secondary 3C5 Keywords. Analytic functions; Univalent functions; Bi-Univalent functions; Differential subordination; Fekete-Segö problem. Received: 4 October 4; Accepted: 9 December 5 Communicated by Dragan S. Djordjević addresses: harimsri@math.uvic.ca H. M. Srivastava), draducanu@unitbv.ro Dorina Răducanu), p.aprawa@pollub.pl Paweł Zaprawa)

2 H. M. Srivastava et al. / Filomat 3:4 6), Let M be the class of analytic functions φ) in U, normalied by φ) =. Also let N be the subclass of M consisting of all univalent functions φ for which φu) is a convex domain. We denote by P the well-known class of analytic functions p) with p) = R p) ) > U). We also denote by B the class of analytic functions ω) in U with ω) = ω) < U). Suppose that the functions f are analytic in U. Then the function f is said to be subordinate to the function, denoted by f, if there exists a function ω B such that f ) = ω) ) U). For functions f given by ) A given by ) = + b n n n= U), the Hadamard product or convolution), denoted by f, is defined by f )) := + a n b n n =: f )) n= U). Recently, Silverman Silvia [5] considered the following classes of functions: { L α = f : f A R f ) + + ) } eiα f ) > U) ) L α b) = { f : f A f ) + + } eiα f ) b < b U), 3) where α π, π] b >. Clearly, if b, then L αb) L α. For each of these two classes of functions, they obtained extreme points, coefficient estimates convolution characteriations. Trojnar-Spelina [3], on the other h, studied the function class LP α given by { LP α = f : f A f ) + + } eiα f ) Q) U), 4) where α π, π]. The function Q) defined by Q) = Q) = + )] + [log π U) 5) maps U onto the domain given by Ω = { w : w C w < Rw) }. Motivated by some of the ideas explored in the aforecited investigations [5] [3], here we define a new class of analytic functions.

3 H. M. Srivastava et al. / Filomat 3:4 6), Definition. Let α π, π] let φ M. A function f A is said to be in the class R α φ) if the following differential subordination is satisfied: f ) + + eiα f ) φ) U). 6) Consider the following two functions: φ ) = + + φ b ) = b ) U) 7) U; b > ). 8) Then it is easy to observe that the corresponding classes R α φ ) R α φ b ) reduce to the classes L α L α b), respectively. We note also that the class R α Q), where the function Q is defined by 5), reduces to function class LP α. We now recall that the function class R given by R = R φ ) = { f : f A R f ) + f ) ) > U) } 9) was investigated by Chichra [7] also by Singh Singh [6]. Another function class R β given by R β = { f : f A R f ) + f ) ) > β U) }, ) which was considered by Silverman [4], can also be obtained from R α φ) upon setting where φ β ) = + β) α = φ = φ β, U; β < ). ) In its special case when β =, the function class R β reduces to the function class R considered by Silverman [4]. In this paper, we investigate various convolution integral characteriations, coefficient estimates subordination results for the general function class R α φ) which we have introduced here by Definition above. In particular, in Section 6, we derive the Fekete-Segö coefficient functional associated with the kth root transform [ f k )] /k of functions in the class R α φ). A similar problem for a corresponding class R Σ;α φ) of bi-univalent functions is also considered in the last section Section 7) of this paper.. Convolution Characteriation In this section we obtain a membership characteriation of the class R α φ) in terms of convolution. Theorem. Let α π, π] let φ M. A necessary sufficient condition for a function f A to be in the class R α φ) is given by f ) + ) e iα ) φe iθ ) U; θ [, π). ) 3 Proof. We have f R α φ) if only if f ) + + eiα f ) φ) U).

4 It follows that f R α φ) if only if H. M. Srivastava et al. / Filomat 3:4 6), Since f ) + + eiα f ) φe iθ ) U; θ [, π) ). we have f ) + + eiα f ) = eiα or, equivalently, f ) = f ) f ) + + eiα f ) ) f ) = f ) ) f ) + + eiα e f iα ) = f ) eiα f ) ) = + + eiα f ) + ) e iα φe iθ ). ) 3 U), ) ) φe iθ ) ) ) The convolution characteriation asserted by Theorem is thus proved. 3. Integral Representation In this section an integral representation for functions in the class R α φ) is provided. Theorem. Let α π, π) let φ M. Suppose also that γ := + e iα. Then f R α φ) if only if there exists ω B such that the following equality: γ η f ) = ζ γ φ ωζ) ) ) dζ dη 3) η γ holds true for all U. Proof. It follows from Definition of the function class R α φ) that f R α φ) if only if there exists ω B such that f ) + + eiα f ) = φ ω) ) U). 4) Making use of ) in the above equality 4), we obtain It follows that which is equivalent to e iα ) e iα + e iα f ) + + eiα f ) ) ) = φ ω) U). f ) + f ) ) = + e iα φ ω) ) U), γ ) γ f ) + γ f ) ) = γ γ φ ω) ) U),

5 H. M. Srivastava et al. / Filomat 3:4 6), where γ =, α π. + eiα We thus find that γ f ) )) = γ γ φ ω) ), which readily yields γ f ) = γ ζ γ φ ωζ) ) dζ. 5) Integrating once more the equality 5), we get 3). The proof of Theorem is thus completed. Remark. If α π, then the equality 4) reduces to f ) = φ ω) ) U). It follows that f R π φ) if only if f ) = φ ωζ) ) dζ. For θ [, π) τ [, ], we now define the function f, θ, τ) by [ γ η ) ] e f, θ, τ) = η γ ζ γ iθ ζζ + τ) φ dζ dτ U). 6) + ζτ By virtue of Theorem, the function f, θ, τ) belongs to the class R α φ). 4. Coefficient Estimates In this section we obtain coefficient estimates for functions belonging to the class R α φ). Theorem 3. Let α π, π] let the function φ) given by φ) = + A + A + be in the class N. If a function f of the form ) belongs to the class R α φ), then A a n n n + + n ) cos α n ). Proof. Since f R α φ), we have where f ) + + eiα f ) = p) U), 7) p) = + p n n φ). n= Equating the coefficients of n on both sides of 7), we find the following relation between the coefficients: n [ + + eiα )n )]a n = p n n ). 8) Since the function φ is univalent in U φu) is a convex domain, we can apply Rogosinski s lemma see []). We thus find that p n A, n.

6 H. M. Srivastava et al. / Filomat 3:4 6), Making use of 8), we get a n which completes the proof of Theorem 3. A n + + eiα )n ) = A n n + + n ) cos α, Remark. i) Let φ) = φ ) defined by 7). If f of the form ) is in the class R α φ ) = L α, then by Theorem 3, we obtain the coefficient estimates found in [5], namely a n n n + + n ) cos α n ). If, in the above inequality, we set α π, then we get a n n n ), which is the well-known coefficient estimates for the class R see [] [9]). ii) Let φ) = Q) defined by 5) let f of the form ) be in the class R α Q) = LP α. Since it follows from Theorem 3, that a n which is the same with the inequality found in [3]. Q) = + 8 π +, 8 nπ n + + n ) cos α = 8 nπ + n + eiα ), n 5. Results Involving Differential Subordination In order to prove our main results of this section, we need the following lemma due to Hallenbeck Ruscheweyh []. Lemma. see []) Let h be a convex function with h) = a let γ C with Rγ. If the function p) given by p) = a + p n n + p n+ n+ + is analytic in U p) + γ p ) h) U), 9) then p) q) h) U), ) where q) = γ n γ/n hζ)ζ γ/n dζ. ) The result is sharp.

7 H. M. Srivastava et al. / Filomat 3:4 6), Theorem 4. Let α π, π) let φ N. If f R α φ), then f ) f ) φt /γ )dt φ) U) ) φrt /γ )dr dt U), 3) where The results are sharp. γ = + e iα. Proof. Assume that f R α φ). Then, from Definition, it follows that the differential subordination 6) holds true. Let p) = f ). Also let γ = + e. iα Then p) + γ p ) = f ) + + eiα f ) φ) U). Since φ N Rγ) for α π, π), in view of Lemma, we have p) γ γ ζ γ φζ)dζ φ) U). 4) With the substitution ζ = t /γ in the integral in 4), by taking into account the fact that p) = f ), the differential chain 4) yields f ) φt /γ )dt φ). The first condition ) of Theorem 4 is thus proved. In order to obtain the differential subordination 3), we show that the function h) given by h) = φt /γ )dt U) 5) belongs to the class N. To prove this, we employ the same technique as in []. We first define Φ γ ) = dt = t/γ n= γ n + γ n. 6) For Rγ) >, the function Φ γ ) is convex in U see [3]). From 6) we obtain φ) Φ γ ) = dt φ) = φt /γ )dt = h). t/γ It was proved in [] that the convolution of two convex functions is also convex. Therefore, the function h) defined by 5) is convex in U. Moreover, since h) =, it follows that h N. We now let p) = f ).

8 Then, by making use of ) 5), we have H. M. Srivastava et al. / Filomat 3:4 6), p) + p ) = f ) φt /γ )dt = h) U). By applying Lemma once more with γ =, we obtain p) hζ)dζ h) U). 7) With the substitution ζ = r in the integral in 7), if we take into account 5) also that the first differential subordination in 7) implies that f ) p) = f ), φ rt /γ) dr dt. The differential subordination 3) is thus proved. Since the result in Lemma is sharp, it follows that the differential subordinations in ) 3) are also sharp. Consequently, the proof of Theorem 4 is completed. The next result is an immediate consequence of Theorem 4. Corollary. Let f be in the class R β β < ) defined by ). Then f ) β β) log ) U) Consider the function f ) β β) log r)dr r φ M ) = + M M > ) the corresponding function class R α φ M ) given by R α φ M ) = { U). f : f A f ) + + } eiα f ) M U; M > ). Variations of the class R φ M ) have been investigated in several works see, for example, [36] []). The following result is another consequence of Theorem 4. Corollary. Let the function f be in the class R α φ M ). Then f ) M cos α U) f ) M cos α U; π < α < π).

9 H. M. Srivastava et al. / Filomat 3:4 6), The Fekete-Segö Problem for the Function Class R α φ) The problem of finding sharp upper bounds for the coefficient functional a 3 µa for different subclasses of the normalied analytic function class A is known as the Fekete-Segö problem. Over the years, this problem has been investigated by many works including for example) [8], [3], [6], [8], [9], [8] [9],. In this section in the next one, it will be assumed that the function φ) is a member of the class M has positive real part in U. Since φ M, its Taylor-Maclaurin series expansion is of the form: φ) = + A + A + U). 8) Remark 3. In view of ), if f R α φ) α π, π), then f ) φ), which when combined with R φ) ) > implies that R f ) ) >. When α π, the class R π φ) consists of all functions f satisfying the same subordination f ) φ). The well-known Noshiro-Warschawski theorem see [9] []) states that a function f A with R f ) ) > is univalent in U. Therefore, for all α π, π], R α φ) is a class of univalent functions, that is, R α φ) is a subclass of the normalied univalent function class S. Recently, Ali et al. [] considered the Fekete-Segö functional associated with the kth root transform for several subclasses of univalent functions. We recall here that, for a univalent function f ) of the form ), the kth root transform is defined by F) = [ f k ) ] /k = + b kn+ kn+ U). 9) n= In view of Remark 3, the functions in the class R α φ) are univalent. Therefore, following the same method as in [], we consider the problem of finding sharp upper bounds for the Fekete-Segö coefficient functional associated with the kth root transform for functions in the class R α φ). Lemma below is needed to prove our main result. Lemma see [4]). Let the function p) given by be in the class P. Then, for any complex number s, p) = + p + p + p sp max {, s }. 3) The result is sharp for the function p) given by p) = + or p) = +. Theorem 5. Let α π, π] let φ M be given by 8). Suppose also that the function f of the form ) is a member of the class R α φ) the function F is the kth root transform of f defined by 9). Then, for any complex number µ, { b k+ µb k+ A 3k cos α max, A µ + k ) 3 + eiα )A } A k3 + e iα ). 3) The result is sharp.

10 H. M. Srivastava et al. / Filomat 3:4 6), Proof. Let f R α φ). Then, clearly, there exists ω B such that f ) + + eiα f ) = φ ω) ). 3) We now define p) = + ω) ω) = + p + p +. 33) Since ω B, it follows that p P. We thus find from 33) that ω) = p + p ) p +. 34) Combining 8) 34), we have φ ω) ) = + A p + 4 A p + A p ) p ) +. Equating the coefficients of on both sides of 3), we get a = A p 3 + e iα ) 35) a 3 = 3 + e iα ) 4 A p + A p ) p ). 36) For f given by ), a computation shows that F) = [ f k )] /k = + k a k+ + k a 3 ) k a k k ) The equations 9) 37) lead us to b k+ = k a b k+ = k a 3 k a k. 38) Substituting from 35) 36) into 38), we obtain so that Let b k+ = b k+ = A p k3 + e iα ) 3k + e iα ) 4 A p + A p ) p ) k )A p 8k 3 + e iα ), [ b k+ µb k+ = A p 6k + e iα A + µ + k ) 3 + ) ] eiα )A p ) A k3 + e iα ). s = A + µ + k ) 3 + ) eiα )A. A k3 + e iα ) The inequality 3) now follows as an application of Lemma. It is easy to check that the result is sharp for the kth root transforms of the functions f, θ, ) f, θ, ) defined by 6) with τ = τ =, respectively. This evidently completes our proof of Theorem 5.

11 H. M. Srivastava et al. / Filomat 3:4 6), For k =, the kth root transform of f reduces to the function f itself. Corollary 3 below is an immediate consequence of Theorem 5. Corollary 3. Let α π, π] let the function φ M be given by 8). Suppose also that the function f of the form ) is in the class R α φ). Then, for any complex number µ, { a 3 µa A cos α max, A µ 3 + eiα )A } A 3 + e iα ). The result is sharp. 7. The Fekete-Segö Problem for the Bi-Univalent Function Class R Σ;α φ) The famous Koebe one-quarter theorem see [9]) ensures that the image of the open unit disk U under every univalent function f A contains a disk of radius. Consequently, every univalent function f has 4 an inverse f satisfying the following relationships: f f ) ) = U) f f w) ) = w w < r f ); r f ) ). 4 In some cases, the inverse function f can be extended to the whole disk U, in which case f is also univalent in U. A function f A is said to be bi-univalent in U if both f f are univalent in U. It is easy to check that a bi-univalent function f given by ) has the inverse f with the series expansion of the form: f w) = w a w + a a 3)w ) Lewin [5] considered the class Σ of bi-univalent functions obtained the bound for the second coefficient. Netanyahu [7] Brannan et al. see [6] [5]) subsequently studied similar problems in this direction. The paper of Srivastava et al. [3] has revived the study of bi-univalent functions in recent years. It was followed by a great number of papers on this topic see, for example, [3], [4], [], [7], [33], [3] [34]). In view of Remark 3, the functions in the class R α φ) are univalent. This motivates the next definition of the class R Σ;α φ). Definition. A function f Σ is said to be in the class R Σ;α φ) if the following subordination relationships hold true: where f ) + + eiα f ) φ) w) + + eiα w w) φw), 4) w) = f w). We find from Definition that, if f R Σ;α φ), then both f = f are univalent in U. For this reason we can consider their corresponding kth root transforms F) = [ f k )] /k

12 H. M. Srivastava et al. / Filomat 3:4 6), given by 9) Gw) = [ w k )] /k = w + d kn+ w kn+ w) = f w) ). 4) n= In this section we derive upper bounds for the Fekete-Segö functional associated with the kth root transform of functions in the class R Σ;α φ). Theorem 6. Let α π, π] let φ M be given by 8). Suppose also that the function f of the form ) is in the class R Σ;α φ) F is the kth root transform of f defined by 9). Then, for any real number µ, b k+ µb k+ A 3k cos α A 3 k+ µ k 3 + e iα )A eiα ) A A ) k+ µ k + A A 3+e iα ) ) 3A +e iα k+ µ k + A A 3+e iα ) ) 3A +e iα 4) Proof. Let f R Σ;α φ). Then, in view of 4), we obtain f ) + + eiα f ) = φ u) ) 43) w) + + eiα w w) = φ vw) ), 44) where u, v B. Suppose that w) = w + c n w n. n= 45) Define As in the proof of Theorem 5, we have p, q P p) = + u) u) = + p + p + U) q) = + v) v) = + q + q + U). φ u) ) = + A p + 4 A p + A p ) p ) + φ v) ) = + A q + 4 A q + A q ) q ) +. It follows from 43) that a = A p 3 + e iα ) a 3 = 3 + e iα ) 4 A p + A p ) p ). 46)

13 H. M. Srivastava et al. / Filomat 3:4 6), Moreover, the equality 44) in conjunction with 45) yields Since c = A q 3 + e iα ) c 3 = Gw) = [ w k )] /k = w + k c w k+ + it follows from 4) 45) that 3 + e iα ) 4 A q + A q ) q ). 47) k c 3 ) k c k w k+ + 48) d k+ = k c d k+ = k c 3 k c k. 49) On the other h, from 39) 45) we get c = a c 3 = a a 3. 5) The equalities 49) 5) give d k+ = k a d k+ = k a a 3) k a k. 5) Furthermore, from 38) 5), we have d k+ = b k+ d k+ = k + )b k+ b k+. 5) Combining the equalities 38), 46), 47), 49) 5), after some simple calculations, we obtain 3 + e iα )kb k+ = A p, 53) 3 + e iα )kb k+ = 4 A p + A p p ) k 3 + e iα ) 8k 3 + e iα ) A p, 54) 3 + e iα )kb k+ = A q 55) 3 + e iα )k[k + )b k+ b k+] = 4 A q + A q q ) k 3 + e iα ) 8k 3 + e iα ) A q. 56) Now, in order to prove the inequality 4), we apply the same technique as in [35]. Indeed, from 53) 55), we get p = q. 57) Substracting 56) from 54) using 57), we have b k+ = k + b k+ + A p q ) k + e iα ). 58) Moreover, the sum between 54) 56) gives 3 + e iα )kk + )b k+ = A p + q ) + A A )p k 3 + e iα ) 4k 3 + e iα ) A p,

14 which, in conjunction with 53), yields H. M. Srivastava et al. / Filomat 3:4 6), b k+ = A 3 p + q ) 4k [3 + e iα )A eiα ) A A )]. 59) On the other h, from 58) 59), we obtain where b k+ µb k+ = A [ p ht) + ) + q k + e iα ht) ) ], ) ht) = 3A + eiα ) k+ µ ) k[3 + e iα )A eiα ) A A )]. Since the functions p q are in the class P, it follows that see [9]) Therefore, we have b k+ µb k+ A 3k + e iα A ht) 3k + e iα which completes the proof of Theorem 6. p q. ht) ) ht) ), Since, for k =, the kth root transform reduces to the function itself, the next result is an immediate consequence of Theorem 6. Corollary 4. Let α π, π] let φ M be given by 8). Suppose also that the function f of the form ) belongs to the class R Σ;α φ). Then, for any real number µ, a 3 µa A cos α A 3 µ 3 + e iα )A eiα ) A A ) µ µ + A A 3A + A A 3A 3+e iα ) +e iα ) 3+e iα ) +e iα ) Finally, when α π, the inequality 6) reduces to a result obtained by Zaprawa [35]. 6) 8. Concluding Remarks Observations In our present investigation, we have successfully applied the principle of differential subordination between analytic functions. Indeed, for α π, π], we have considered a certain function class R α φ) of all normalied analytic functions in the open unit disk U, which satisfy the following differential subordination: f ) + ) + e iα f ) φ) U), where the function φ) is analytic in U such that φ) =. In particular, we have investigated various integral convolution characteriations, coefficient estimates differential subordination results for functions belonging to the class R α φ). We have also derived the Fekete-Segö coefficient functional associated with the kth root transform [ f k )] /k of functions in R α φ). Furthermore, we have considered a similar problem for a corresponding class R Σ;α φ) of bi-univalent functions. We have pointed out relevant connections of the results presented here with previous known results.

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