An application of Schur algorithm to variability regions of certain analytic functions-II
aa r X i v : . [ m a t h . C V ] J un AN APPLICATION OF SCHUR ALGORITHM TO VARIABILITYREGIONS OF CERTAIN ANALYTIC FUNCTIONS-II
MD FIROZ ALI, VASUDEVARAO ALLU, AND HIROSHI YANAGIHARA
Abstract.
We continue our study on variability regions in [2], where the authorsdetermined the region of variability V j Ω ( z , c ) = { R z z j ( g ( z ) − g (0)) dz : g ( D ) ⊂ Ω , ( P − ◦ g )( z ) = c + c z + · · · + c n z n + · · · } for each fixed z ∈ D , j = − , , , , . . . and c = ( c , c , . . . , c n ) ∈ C n +1 , when Ω ( C is a convex domain, and P is aconformal map of the unit disk D onto Ω . In the present article, we first show thatin the case n = 0 , j = − and c = 0 , the result obtained in [2] still holds when oneassumes only that Ω is starlike with respect to P (0) . Let CV (Ω) be the class ofanalytic functions f in D with f (0) = f ′ (0) − satisfying zf ′′ ( z ) /f ′ ( z ) ∈ Ω .As applications we determine variability regions of log f ′ ( z ) when f ranges over CV (Ω) with or without the conditions f ′′ (0) = λ and f ′′′ (0) = µ . Here λ and µ are arbitrarily preassigned values. By choosing particular Ω , we obtain the precisevariability regions of log f ′ ( z ) for other well-known subclasses of analytic andunivalent functions. Introduction
Let C be the complex plane. For c ∈ C and r > , let D ( c, r ) := { z ∈ C : | z − c |
Mathematics Subject Classification.
Primary 30C45, 30C75.
Key words and phrases.
Analytic functions, univalent functions, convex functions, starlike func-tions, Banach space, norm, Schur algorithm, subordination, variability region. { log f ′ ( z ) : f ∈ CV } = (cid:26) log 1(1 − z ) : | z | ≤ | z | (cid:27) . For f ∈ CV , an easy consequence of the Schwarz’s lemma is that that | f ′′ (0) | ≤ .For fixed z ∈ D and λ ∈ D , Gronwall [6] obtained the sharp lower and upperestimates for | f ′ ( z ) | , when f ∈ CV satisfies the additional condition f ′′ (0) = 2 λ (see also [5]). Let e V ( z , λ ) = { log f ′ ( z ) : f ∈ CV and f ′′ (0) = 2 λ } . If | λ | = 1 , then by Schwarz’s lemma, for f ∈ CV the condition f ′′ (0) = 2 λ forces f ( z ) ≡ z/ (1 − λz ) , and hence e V ( z , λ ) = { log 1 / (1 − λz ) } . Since e V ( e − iθ z , e iθ λ ) = e V ( z , λ ) for all θ ∈ R , without loss of generality we may assume that ≤ λ < . In2006, Yanagihra [20] obtained the following extension of Gronwall’s [6] result. Theorem A.
For any z ∈ D \{ } and ≤ λ < the set e V ( z , λ ) is a convex closedJordan domain surrounded by the curve ( − π, π ] ∋ θ − λ cos( θ/ p − λ sin ( θ/ ! log ( − e iθ/ z iλ sin( θ/ − p − λ sin ( θ/ ) − λ cos( θ/ p − λ sin ( θ/ ! log ( − e iθ/ z iλ sin( θ/
2) + p − λ sin ( θ/ ) . Theorem A, can be equivalently written as follows.
Theorem B.
Let H = { w ∈ C : Re w > } . For any z ∈ D \{ } and ≤ λ < ,the variability region (cid:26)Z z g ( ζ ) − g (0) ζ dζ : g ∈ A ( D ) with g (0) = 1 , g ′ (0) = 2 λ, g ( D ) ⊂ H (cid:27) coincides with the same convex closed Jordan domain as in Theorem A. Theorem A is a direct consequence of Theorem B if we choose g ( z ) = 1 + zf ′′ ( z ) /f ′ ( z ) . For similar results, we refer to [12, 13, 18, 19, 21] and the referencestherein.Recently, the present authors [2] extended Theorem B to the most general setting.Let Ω be a simply connected domain in C with Ω = C , and P be a conformalmap of D onto Ω . Let F Ω be the class of analytic functions g in D with g ( D ) ⊂ Ω .Then the map P − ◦ g maps D into D . For c = ( c , c , . . . , c n ) ∈ C n +1 , let F Ω ( c ) = { g ∈ F Ω : ( P − ◦ g )( z ) = c + c z + · · · + c n z n + · · · in D } . ariability regions of certain analytic and univalent functions 3 Let H ∞ ( D ) be the Banach space of analytic functions f in D with the norm k f k ∞ =sup z ∈ D | f ( z ) | , and H ∞ ( D ) be the closed unit ball of H ∞ ( D ) , i.e., H ∞ ( D ) = { ω ∈ H ∞ ( D ) : k ω k ∞ ≤ } . We note that the coefficient body C ( n ) defined by C ( n ) = { c = ( c , c , . . . , c n ) ∈ C n +1 : where there exists ω ∈ H ∞ ( D ) such that ω ( z ) = c + c z + · · · + c n z n + · · · in D } is a compact and convex subset of C n +1 . The coefficient body C ( n ) has been com-pletely characterized Schur [15, 16]. For a detailed treatment, we refer to [7, ChapterI] and [3, Chapter 1].We call c = ( c , . . . , c n ) the Carathéodory data of length n + 1 . For a givenCarathéodory data c = ( c , . . . , c n ) ∈ C n +1 , the Schur parameter γ = ( γ , . . . , γ k ) , k = 0 , , . . . , n is defined as follows.For j = 0 , , . . . , define recursively c ( j ) = ( c ( j )0 , c ( j )1 . . . , c ( j ) n − j ) and γ j = c ( j )0 by(1.1) c ( j )0 = c ( j − − | γ j − | , c ( j ) p = c ( j − p +1 + γ j − P pℓ =1 c ( j ) p − ℓ c ( j − ℓ − | γ j − | (1 ≤ p ≤ n − j ) , with c (0) = c = ( c , . . . , c n ) . In the j th step ( j = 0 , , . . . ), if | γ j | > , then we put k = j and γ = ( γ , . . . , γ j ) ; if | γ j | = 1 , then we put k = n , and for p = j + 1 , . . . , n ,we take γ p = ∞ , if c ( j ) p − j = 00 , if c ( j ) p − j = 0; if | γ j | < then we proceed to ( j + 1) th step and so on. Applying this procedurerecursively we obtain the Schur parameter γ = ( γ , . . . , γ k ) , k = 0 , . . . , n of c =( c , . . . , c n ) .When | γ | < , . . . , | γ n | < , c = ( c , . . . , c n ) = c (0) and γ = ( γ , . . . , γ n ) aredetermined uniquely each other. For an explicit representation of γ in terms of c ,we refer to [15, 16]. For given c = ( c , . . . , c n ) ∈ C n +1 , Schur [15, 16] proved that c ∈ Int C ( n ) , c ∈ ∂ C ( n ) and c
6∈ C ( n ) are respectively equivalent to the conditions ( C1 ) k = n and | γ i | < for i = 1 , , . . . , n ( C2 ) , k = n and | γ | < , . . . , | γ i − | < , | γ i | = 1 , γ i +1 = · · · = γ n = 0 for some i = 0 , . . . , n and ( C3 ) the hypothesesthat either ( C1 ) or ( C2 ) does not hold. Furthermore, for c ∈ Int C ( n ) , the Schurparameter can be computed as follows.Let ω ∈ H ∞ ( D ) be such that ω ( z ) = c + c z + · · · + c n z n + · · · . Define ω ( z ) = ω ( z ) and ω k ( z ) = ω k − ( z ) − ω k − (0) z (1 − ω k − (0) ω k − ( z )) ( k = 1 , , . . . , n ) . Then γ p = ω p (0) , ω p ( z ) = c ( p )0 + c ( p )1 z + · · · + c ( p ) n − p z n − p + · · · hold for p = 0 , , . . . , n . For a detailed proof, we refer to [7, Chapter 1].For a ∈ D , define σ a ∈ Aut ( D ) by σ a ( z ) = z + a az , z ∈ D . Md Firoz Ali, Vasudevarao Allu and Hiroshi Yanagihara
For ε ∈ D and the Schur parameter γ = ( γ , . . . , γ n ) of c ∈ Int C ( n ) , let ω γ,ε ( z ) = σ γ ( zσ γ ( · · · zσ γ n ( εz ) · · · )) , z ∈ D , (1.2) Q γ,j ( z, ε ) = Z z ζ j { P ( ω γ,ε ( ζ )) − P ( c ) } dζ , z ∈ D and ε ∈ D . (1.3)Then ω γ,ε ∈ H ∞ ( D ) with Carathéodory data c , i.e., ω γ,ε ( z ) = c + c z + · · · + c n z n + · · · . By using the Schur algorithm, recently the present authors [2] obtained thefollowing general result for the region of variability. Theorem C. [2]
Let n ∈ N ∪ { } , j ∈ {− , , , , . . . } , and c = ( c , . . . , c n ) ∈ C n +1 be a Carathéodory data. Let Ω be a convex domain in C with Ω = C , and P be aconformal map of D onto Ω . For each fixed z ∈ D \{ } , let V j Ω ( z , c ) = (cid:26)Z z ζ j ( g ( ζ ) − g (0)) dζ : g ∈ F Ω ( c ) (cid:27) . (i) If c = ( c , . . . , c n ) ∈ Int C ( n ) and γ = ( γ , . . . , γ n ) be the Schur parameter of c ,then Q γ,j ( z , ε ) defined by (1.2) is a convex univalent function of ε ∈ D and V j Ω ( z , c ) = Q γ,j ( z , D ) := { Q γ,j ( z , ε ) : ε ∈ D } . Furthermore, Z z ζ j { g ( ζ ) − g (0) } dζ = Q γ,j ( z , ε ) for some g ∈ F Ω ( c ) and ε ∈ ∂ D if, and only if, g ( z ) ≡ P ( ω γ,ε ( z )) . (ii) If c ∈ ∂ C ( n ) and γ = ( γ , . . . , γ i , , . . . , is the Schur parameter of c , then V j Ω ( z , c ) reduces to a set consists of a single point w , where w = Z z ζ j { P ( σ γ ( ζ σ γ ( · · · ζ σ γ i − ( γ i ζ ) · · · ))) − P ( c ) } dζ . (iii) If c
6∈ C ( n ) then V j Ω ( z , c ) = ∅ . In the present article, we first show that in the case n = 0 , j = − and c = 0 , theconclusion of Theorem C holds when one weakens the assumption that Ω is convex tothat of starlikeness of Ω with respect to P (0) . We then present several applicationsof Theorems B and 2.1 to obtain the precise variability region of different quantitiesfor several well-known subclasses of analytic and univalent functions. We also obtaincertain subordination results. 2. Main Results
Theorem 2.1.
Let b ∈ C , z ∈ D \{ } and Ω be a starlike domain with respect to b satisfying Ω = C . Let P be a conformal map of D onto Ω with P (0) = b . Then theregion of variability V − ( z ,
0) = (cid:26)Z z g ( ζ ) − bζ dζ : g ∈ F Ω , g (0) = b (cid:27) is a convex closed Jordan domain, and coincides with the set K ( D (0 , | z | )) , where K ( z ) = R z ζ − ( P ( ζ ) − b ) dζ . Furthermore, for | ε | = 1 and g ∈ F Ω with g (0) = b ,the relation R z ζ − ( g ( ζ ) − b ) dζ = K ( εz ) holds if, and only if, g ( z ) ≡ P ( εz ) . ariability regions of certain analytic and univalent functions 5 Proof.
Let g ∈ A ( D ) be such that g (0) = b and g ( D ) ⊂ Ω . Then g ≺ P , i.e., g issubordinate to P . By using a result of Suffridge [17], we may conclude Z z g ( ζ ) − bζ dζ ≺ K ( z ) := Z z P ( ζ ) − bζ dζ . Thus there exists ω ∈ H ∞ ( D ) with ω (0) = 0 and R z ζ − { g ( ζ ) − b } dζ = K ( ω ( z )) .From this it follows that V − ( z , ⊂ { K ( ω ( z )) : ω ∈ H ∞ ( D ) and ω (0) = 0 } = K (cid:0) D (0 , | z | ) (cid:1) . For ε ∈ D , let g ε ( z ) = P ( εz ) . Then g ε (0) = P (0) = b and g ε ( D ) = P ( D ) = Ω .Therefore K ( εz ) = Z εz P ( ζ ) − bζ dζ = Z z g ε ( ζ ) − bζ dζ ∈ V − ( z , , and hence K ( D (0 , | z | )) ⊂ V − ( z , .We now deal with the uniqueness. Suppose that(2.1) Z z g ( ζ ) − bζ dζ = K ( εz ) holds for some g with g (0) = b and g ( D ) ⊂ Ω , and | ε | = 1 . Then there exists ω ∈ H ∞ ( D ) with ω (0) = 0 such that R z ζ − { g ( ζ ) − b } dζ = K ( ω ( z )) . From (2.1) wehave K ( ω ( z )) = K ( εz ) . Since K is a convex univalent function, ω ( z ) = εz . Itfollows from Schwarz’s lemma that ω ( z ) ≡ εz . Consequently g ( z ) ≡ P ( εz ) . (cid:3) The Class CV (Ω) . Suppose that Ω is a simply connected domain with ∈ Ω .Define CV (Ω) = (cid:26) f ∈ A ( D ) : 1 + z f ′′ ( z ) f ′ ( z ) ∈ Ω for all z ∈ D (cid:27) . Let P be the conformal map of D onto Ω with P (0) = 1 . Then for each f ∈ CV (Ω) ,we have zf ′′ ( z ) /f ′ ( z ) ≺ P . For α ∈ R , let H α := { z ∈ C : Re z > α } and H = H . When Ω = H and P ( z ) = (1 + z ) / (1 − z ) , CV ( H ) = CV is thewell-known class of normalized convex functions in D . If Ω ⊂ H , then CV (Ω) is asubclass of CV . For ≤ α < , CV ( α ) := CV ( H α ) is the class of convex functionsof order α . In this case, P ( z ) = { − α ) z } / (1 − z ) . If < β ≤ , then CV β := CV ( { w ∈ C : | arg w | < πβ/ } ) is the class of strongly convex functions oforder β and P is given by P ( z ) = { (1 + z ) / (1 − z ) } β .As an application of Theorem 2.1 we determine the variability region of log f ′ ( z ) when f ranges over CV (Ω) . Theorem 2.2.
Let Ω be a starlike domain with respect to , and P be a conformalmap of D onto Ω with P (0) = 1 . Then for each fixed z ∈ D \{ } , the region ofvariability V CV (Ω) ( z ) := { log f ′ ( z ) : f ∈ CV (Ω) } is a convex closed Jordan domain, and coincides with the set K ( D (0 , | z | )) , where K ( z ) = R z ζ − ( P ( ζ ) − dζ is a convex univalent function in D . Furthermore, log f ′ ( z ) = K ( εz ) for some | ε | = 1 and f ∈ CV (Ω) if, and only if, f ( z ) = ε − F ( εz ) ,where F ( z ) = R z e K ( ζ ) dζ . Md Firoz Ali, Vasudevarao Allu and Hiroshi Yanagihara
Proof.
Let c = 0 ∈ C be a given Carathéodry data of length one. Then F Ω (0) = { g ∈ A ( D ) : g ( D ) ⊂ Ω and ( P − ◦ g )(0) = 0 } . It is easy to see that the map CV (Ω) ∋ f g ( z ) = 1 + z f ′′ ( z ) f ′ ( z ) ∈ F Ω (0) is bijective. Indeed, since g ( z ) = 1 + zf ′′ ( z ) /f ′ ( z ) is analytic in D , f ′ ( z ) does nothave zeros in D , and so(2.2) log f ′ ( z ) = Z z ζ − ( g ( ζ ) − dζ , where log f ′ is a single valued branch of the logarithm of f ′ with log f ′ (0) = 0 . Theconclusions now follows from Theorem 2.1 and (2.2). (cid:3) As an application of Theorems C we determine the variability region of log f ′ ( z ) ,when f ranges over CV (Ω) with the conditions f ′′ (0) = 2 λ and f ′′′ (0) = 6 µ . Here z ∈ D \{ } , λ, µ ∈ C are arbitrarily preassigned values. By letting Ω be one of theparticular domains mentioned in the above, we can determine variability regions of log f ′ ( z ) for various subclasses of CV .Let Ω be a simply connected domain with Ω = C , and P be a conformal map of D onto Ω with P ( z ) = α + α z + α z + · · · . Let g be an analytic function in D with g ( z ) = b + b z + b z + · · · satisfying g ( D ) ⊂ Ω . For simplicity we assume that P (0) = g (0) , i.e., α = b . Let ω ( z ) = ( P − ◦ g )( z ) = c + c z + c z + · · · , z ∈ D . Then(2.3) c = 0 , c = b α , c = α b − α b α . By Schwarz’s lemma | b | ≤ | α | , with equality if, and only if, g ( z ) = P ( εz ) forsome ε ∈ ∂ D . Let γ = ( γ , γ , γ ) be the Schur parameter of the Carathéodory data c = (0 , c , c ) . Then γ = ω (0) = c = 0 , and ω ( z ) = ω ( z ) z , γ = ω (0) ,ω ( z ) = ω ( z ) − γ z (1 − γ ω ( z )) , γ = ω (0) . A simple computation shows that(2.4) γ = 0 , γ = c = b α , γ = c − | c | = α ( α b − α b ) α ( | α | − | b | ) . For f ∈ CV (Ω) and k ∈ N , let a k ( f ) = f ( k ) (0) /k ! . Let g ( z ) = 1 + zf ′′ ( z ) /f ′ ( z ) =1 + b z + b z + · · · . Then(2.5) b = 2 a ( f ) and b = 6 a ( f ) − a ( f ) . From (2.4) and (2.5), we have(2.6) γ = 0 , γ = 2 a ( f ) α , γ = 2 α { α a ( f ) − α + α ) a ( f ) } α ( | α | − | a ( f ) | ) . ariability regions of certain analytic and univalent functions 7 Let A (2 , Ω) = { a ( f ) : f ∈ CV (Ω) } . Then by the Schwarz lemma, A (2 , Ω) = D (0 , | α | / . For f ∈ CV (Ω) and λ ∈ ∂ A (2 , Ω) , a ( f ) = λ if, and only if, f ( z ) ≡ γ − F ( γ z ) , where γ = 2 λ/α . Byapplying Theorem C with n = 1 and j = − we obtain the following generalizationof Theorem A. Theorem 2.3.
Let Ω be a convex domain with ∈ Ω and P be a conformal map of D onto Ω with P ( z ) = 1 + α z + · · · . For λ ∈ C with | λ | ≤ | α | / and z ∈ D \{ } ,consider the variability region V CV (Ω) ( z , λ ) := { log f ′ ( z ) : f ∈ CV (Ω) with a ( f ) = λ } . (i) If | λ | = | α | / , then V CV (Ω) ( z , λ ) reduces to a set consists of a single point w ,where w = R z ζ − { P ( γ ζ ) − } dζ with γ = 2 λ/α . (ii) If | λ | < | α | / then V CV (Ω) ( z , λ ) = Q γ ( z , D ) , where γ = 2 λ/α and Q γ ( z , ε ) = Z z ζ − (cid:26) P (cid:18) ζ εζ + γ γ εζ (cid:19) − (cid:27) dζ is a convex, univalent and analytic function of ε ∈ D . Furthermore, log f ′ ( z ) = Q γ ( z , ε ) holds for some ε ∈ ∂ D and f ∈ CV (Ω) with a ( f ) = λ , if, and only if, f ( z ) = Z z e Q γ ( ζ,ε ) dζ , z ∈ D . Next let A (3 , Ω) = { ( a ( f ) , a ( f )) ∈ C : f ∈ CV (Ω) } , and for λ, µ ∈ C , let γ := γ ( λ, µ ) and γ := γ ( λ, µ ) be given by(2.7) γ = 2 λα and(2.8) γ = α { α µ − α + α ) λ } α ( | α | − | λ | ) , if | γ | < , , if | γ | = 1 and α µ = 2( α + α ) λ , ∞ , if | γ | = 1 and α µ = 2( α + α ) λ . Then ( λ, µ ) ∈ A (3 , Ω) if, and only if, one of the following conditions holds.(a) | γ ( λ, µ ) | = 1 and γ ( λ, µ ) = 0 .(b) | γ ( λ, µ ) | < and | γ ( λ, µ ) | = 1 .(c) | γ ( λ, µ ) | < and | γ ( λ, µ ) | < .In case (a), for f ∈ CV (Ω) , ( a ( f ) , a ( f )) = ( λ, µ ) holds if, and only if, g ( z ) = P ( γ z ) , i.e., f ( z ) = γ F ( γ z ) , where γ = γ ( λ, µ ) . Similarly, in case (b), for f ∈ CV (Ω) , ( a ( f ) , a ( f )) = ( λ, µ ) holds if, and only if, g ( z ) = P ( zσ γ ( γ z )) , i.e., f ( z ) = Z z exp (cid:20)Z ζ ζ − { P ( ζ σ γ ( γ ζ )) − } dζ (cid:21) dζ . Md Firoz Ali, Vasudevarao Allu and Hiroshi Yanagihara
We note that ( λ, µ ) ∈ ∂ A (3 , Ω) if, and only if, either (a) or (b) holds.Suppose that (c) holds, i.e., ( λ, µ ) ∈ Int A (3 , Ω) . Then for f ∈ CV (Ω) , ( a ( f ) , a ( f )) =( λ, µ ) holds if, and only if, there exists ω ∗ ∈ H ∞ ( D ) such that g ( z ) = 1 + zf ′′ ( z ) f ′ ( z ) = P ( zσ γ ( zσ γ ( zω ∗ ( z )))) . Let(2.9) Q γ ,γ ( z, ε ) = Z z ζ − { P ( ζ σ γ ( ζ σ γ ( εζ ))) − } dζ , z ∈ D and ε ∈ D . Then for any fixed ε ∈ D , Q γ ,γ ( z, ε ) is an analytic function of z ∈ D , and for eachfixed z ∈ D , Q γ ,γ ( z, ε ) is an analytic function of ε ∈ D . Thus by Theorem C wehave the following. Theorem 2.4.
Let Ω be a convex domain with ∈ Ω and P be a conformal mapof D onto Ω with P ( z ) = 1 + α z + · · · . Let ( λ, µ ) ∈ C and γ = γ ( λ, µ ) and γ = γ ( λ, µ ) defined by (2.7) and (2.8) respectively. For z ∈ D \{ } , consider thevariability region V CV (Ω) ( z , λ, µ ) := { log f ′ ( z ) : f ∈ CV (Ω) with ( a ( f ) , a ( f )) = ( λ, µ ) } . (i) If | γ ( λ, µ ) | = 1 and | γ ( λ, µ ) | = 0 , then V CV (Ω) ( z , λ, µ ) reduces to a set con-sists of a single point w , where w = R z ζ − { P ( γ ζ ) − } dζ . (ii) If | γ ( λ, µ ) | < and | γ ( λ, µ ) | = 1 , then V CV (Ω) ( z , λ, µ ) reduces to a set con-sists of a single point w , where w = R z ζ − { P ( ζ σ γ ( γ ζ )) − } dζ . (iii) If | γ ( λ, µ ) | < and | γ ( λ, µ ) | < , i.e., ( λ, µ ) ∈ Int A (3 , Ω) then Q γ ,γ ( z , ε ) defined by (2.9) is a convex, univalent analytic function of ε ∈ D and V CV (Ω) ( z , λ, µ ) = Q γ ,γ ( z , D ) . Furthermore, log f ′ ( z ) = Q γ ,γ ( z , ε ) for some | ε | = 1 and f ∈ CV (Ω) , with ( a ( f ) , a ( f )) = ( λ, µ ) , if, and only if, f ( z ) = Z z exp (cid:20)Z ζ ζ − { P ( zσ γ ( zσ γ ( εζ ))) − } dζ (cid:21) dζ . Remark 2.1.
For a simply connected domain Ω with ∈ Ω , define S ∗ (Ω) = (cid:26) f ∈ A ( D ) : z f ′ ( z ) f ( z ) ∈ Ω in D (cid:27) . Then f ∈ CV (Ω) if, and only if, zf ′ ( z ) ∈ S ∗ (Ω) . Thus we can easily translate thetheorems of this section to results about variability regions of log { f ( z ) /z } when f ranges over S ∗ (Ω) with or without the conditions f ′′ (0) = λ and f ′′′ (0) = µ . ariability regions of certain analytic and univalent functions 9 Uniformly Convex Functions.
For ≤ k < ∞ , the class k - U CV of k -uniformly convex functions is defined by CV (Ω k ) , where Ω k := { w ∈ C : Re w >k | w − |} . Notice that Ω k is the convex domain containing , bounded by the conicsection. In this case, the conformal map P k which maps the unit disk D conformallyonto Ω k is given by P k = − k cosh (cid:16) A log √ z −√ z (cid:17) − k − k for ≤ k <
11 + π (cid:16) log √ z −√ z (cid:17) for k = 1 k − sin (cid:18) π K ( x ) R u ( z ) / √ x dt √ (1 − t )(1 − x t ) (cid:19) + k k − for < k < ∞ , where A = (2 /π ) arccos k , u ( z ) = ( z − √ x ) / (1 − √ xz ) , and K ( x ) is the ellipticalintegral defined by K ( x ) = Z dt p (1 − t )(1 − x t ) , x ∈ (0 , . For more details concerning uniformly convex functions, we refer to [10] and [14].When k = 0 , the class - U CV is essentially the same as CV . Let P k ( z ) = 1 + α k z + α k z + · · · . Then it is a simple exercise to see that α k = A − k for ≤ k < /π for k = 1 π k − K ( x )(1 + x ) √ x for < k < ∞ . Let f ∈ k - U CV be of the form f ( z ) = z + a z + a z + · · · and g ( z ) = 1+ zf ′′ ( z ) /f ′ ( z ) .Then from (2.3) and (2.5) we obtain | a | ≤ α k / . For z ∈ D \{ } and | λ | ≤ α k / ,consider the region of variability V k - UCV ( z , λ ) = { log f ′ ( z ) : f ∈ k - U CV with a ( f ) = λ } . The following corollary is a simple consequence of Theorem 2.3.
Corollary 2.1.
Let z ∈ D \{ } and λ ∈ C be such that | λ | ≤ α k / . Also let γ = 2 λ/α k . (i) If | γ | = 1 then V k - UCV ( z , λ ) = { w } , where w = R z ζ − { P k ( γ ζ ) − } dζ . (ii) If | γ | < then V k - UCV ( z , λ ) = Q γ ( z , D ) , where Q γ ( z , ε ) = Z z ζ − (cid:26) P k (cid:18) ζ εζ + γ γ εζ (cid:19) − (cid:27) dζ is a convex, univalent and analytic function of ε ∈ D . Furthermore log f ′ ( z ) = Q γ ( z , ε ) for some ε ∈ ∂ D and f ∈ k - U CV with a ( f ) = λ , if, and only if, f ( z ) = Z z e Q γ ( ζ,ε ) dζ , z ∈ D . Janowski Starlike and Convex Function.
For
A, B ∈ C with | B | ≤ and A = B , let P A,B ( z ) := (1 + Az ) / (1 + Bz ) . Then P A,B is a conformal map of D ontoa convex domain Ω A,B . In this case, the classes S ∗ (Ω A,B ) and CV (Ω A,B ) reduces to S ∗ ( A, B ) := (cid:26) f ∈ A ( D ) : zf ′ ( z ) f ( z ) ≺ Az Bz (cid:27) and CV ( A, B ) := (cid:26) f ∈ A ( D ) : zf ′′ ( z ) f ′ ( z ) + 1 ≺ Az Bz (cid:27) respectively. Since P A,B ( D ) = P − A, − B ( D ) , with out loss of generality we may assumethat A ∈ C with − ≤ B ≤ and A = B . It is important to note that functionsin S ∗ ( A, B ) with A ∈ C , − ≤ B ≤ and A = B are not in general univalent. For − ≤ B < A ≤ , it is easy to see that Ω A,B ⊂ H , and so S ∗ ( A, B ) ⊂ S ∗ . Similarresult holds for CV ( A, B ) . (Note that for − ≤ B < A ≤ , the class S ∗ ( A, B ) wasfirst introduced and investigated by Janowski [9]).We also note that P A,B ( z ) := (1 + Az ) / (1 + Bz ) = 1 + ( A − B ) z + · · · . For f ∈ CV ( A, B ) , from (2.3) and (2.5) we immediately obtain | a ( f ) | ≤ | A − B | / . For z ∈ D \{ } and | λ | ≤ | A − B | / , consider the following V CV ( A,B ) ( z ) := { log f ′ ( z ) : f ∈ CV ( A, B ) } ,V CV ( A,B ) ( z , λ ) := { log f ′ ( z ) : f ∈ CV ( A, B ) with a ( f ) = λ } . The following corrolary is a simple consequence of Theorems 2.2 and 2.3.
Corollary 2.2.
Let z ∈ D \{ } be fixed and λ ∈ C be such that | λ | ≤ | A − B | / .Also let γ = 2 λ/ ( A − B ) . (i) The region of variability V CV ( A,B ) ( z ) is a convex, closed, Jordan domain andcoincides with the set K ( D (0 , | z | )) , where K ( z ) = Z z A − B Bζ dζ is a convex, univalent function in D . Furthermore, log f ′ ( z ) = K ( εz ) forsome | ε | = 1 and f ∈ CV ( A, B ) if, and only if, f ( z ) = ε − F ( εz ) , where F ( z ) = R z e K ( ζ ) dζ . (ii) If | γ | = 1 then V CV ( A,B ) ( z , λ ) = { w } , where w = R z ζ − { P ( γ ζ ) − } dζ . (iii) If | γ | < then V CV ( A,B ) ( z , λ ) = Q γ ( z , D ) , where Q γ ( z , ε ) = Z z ( A − B ) σ γ ( εζ )1 + Bζ σ γ ( εζ ) dζ is a convex, univalent and analytic function of ε ∈ D . Furthermore log f ′ ( z ) = Q γ ( z , ε ) holds for some ε ∈ ∂ D and f ∈ CV (Ω) with a ( f ) = λ , if, and only if, f ( z ) = Z z e Q γ ( ζ,ε ) dζ , z ∈ D . ariability regions of certain analytic and univalent functions 11 Remark 2.2.
The region of variability V CV ( A,B ) ( z , λ ) for the class CV ( A, B ) wasfirst obtained by Ul-Haq [18] for − ≤ B < and A > B . Although, Ul-Haqconsidered the problem for A ∈ C , < B ≤ and A = B , it is evident from thecomputation that it is valid only for − ≤ B < and A > B . We also note thatthe Herglotz representation (2) in [18] for functions in CV ( A, B ) is not valid when − < B < , and this was used by by Ul-Haq [18] in order to obtain the region ofvariability V CV ( A,B ) ( z ) .In particular, for A = e − iα with α ∈ ( − π/ , π/ and B = − , the class CV ( A, B ) reduces to the class of functions which satisfy Re { e iα (1 + zf ′′ ( z ) /f ′ ( z )) } > for z ∈ D . This class, denoted by S α are known as Robertson functions. It is interestingto note that if we choose A = e − iα with α ∈ ( − π/ , π/ and B = − in Corollary2.2. then we obtain the result obtained in [13].For A = 1 − α with − / ≤ α < and B = − , the class CV ( A, B ) reduces tothe class of functions f satisfying Re (1 + zf ′′ ( z ) /f ′ ( z )) > α for z ∈ D . This class,denoted by CV ( α ) are known as convex functions of order α . It is important to notethat for ≤ α < , CV ( α ) ⊂ CV . On the other hand, for − / ≤ α < functionsin CV ( α ) are convex functions in some direction (see [11]). If we choose A = 1 − α with − / ≤ α < and B = − in Corollary 2.2 then we obtain the precise regionof variability V CV ( α ) ( z ) := { log f ′ ( z ) : f ∈ CV ( α ) } and V CV ( α ) ( z , λ ) := { log f ′ ( z ) : f ∈ CV ( α ) and a ( f ) = λ } , which also gives a generalization of Theorem A. Inparticular, if we choose A = 2 and B = − in Corollary 2.2 then we obtain theresult obtained by Ponnusamy and Vasudevarao [12, Theorem 2.6]. Similarly, for A = − and B = − , the class CV ( A, B ) reduces to the class of functions f whichsatisfy Re (1 + zf ′′ ( z ) /f ′ ( z )) < / for z ∈ D . Note that functions in the class CV ( − , − are starlike, but not necessarily convex ( [1]), and if we choose A = − and B = − in Corollary 2.2 then we obtain the result obtained in [12, Theorem2.8].Since f ∈ CV ( A, B ) if, and only if, zf ′ ( z ) ∈ S ∗ ( A, B ) , we can easily translate theabove results about variability regions of log { f ( z ) /z } when f ranges over S ∗ ( A, B ) with or without the condition f ′′ (0) = 2 λ .3. Concluding Remark
Theorem 2.1 demonstrates that our results are closely related to the concept ofsubordination. Our assumption g ∈ F Ω ( c ) in Theorem C can be rewritten as g ≺ P when c = 0 . In this case P − ( g ( z )) = c z + · · · + c n z n + · · · . However, apart from afew exceptional cases, we cannot express our conclusions in terms of subordinationrelations. Let c = ( c , . . . , c n − ) = (0 , . . . , ∈ C n . Then the Schur parameter for c is given by γ = ( γ , . . . , γ n − ) = (0 , . . . , . For this particular choice of c , thefunction Q γ,j defined by (1.3) becomes Q γ,j ( z, ε ) = Z z ζ j { P ( εζ n ) − } dζ . Let H ( z ) = j + 1 z ( j +1) /n Z z /n ζ j { P ( ζ n ) − } dζ . Then j + 1 z j +1 Q γ,j ( z, ε ) = H ( εz n ) . Since by Theorem C, for each fixed z ∈ D \{ } , Q γ,j ( z, ε ) is a convex univalentfunction of ε ∈ D , and H ( εz n ) is also a convex univalent function of ε ∈ D . Byletting z → in D , H ( ε ) is also convex univalent in D . Let g ∈ F Ω with g ′ (0) = · · · = g ( n − (0) = 0 . It follows from Theorem C that for any z ∈ D \{ } , there exists ε ∈ D satisfying Z z ζ j { g ( ζ ) − } dζ = Q γ,j ( z, ε ) . Thus for all z ∈ D , we have j + 1 z j +1 Z z ζ j { g ( ζ ) − } dζ = j + 1 z j +1 Q γ,j ( z, ε ) = H ( εz n ) ⊂ H ( D ) . Consequently, in view of univalence of H we obtain the following subordinationrelation j + 1 z j +1 Z z ζ j { g ( ζ ) − } dζ ≺ H ( z ) . This was previously proved by Hallenbeck and Ruscheweyh [8]. In fact, Hallenbeckand Ruscheweyh [8] proved the above subordination relation when Re j ≥ − with j = − . Data availability statement:
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Comput. Methods Funct.Theory , (2010), 291–302. Md Firoz Ali, Department of Mathematics NIT Durgapur, Mahatma Gandhi Av-enue, Durgapur - 713209, West Bengal, India.
E-mail address : [email protected] Vasudevarao Allu, Discipline of Mathematics, School of Basic Sciences, IndianInstitute of Technology Bhubaneswar, Argul, Bhubaneswar, PIN-752050, Odisha(State), India.
E-mail address : [email protected] Hiroshi Yanagihara, Department of Applied Science, Faculty of Engineering, Ya-maguchi University, Tokiwadai, Ube 755, Japan
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