Bohr phenomenon for certain close-to-convex analytic functions
aa r X i v : . [ m a t h . C V ] S e p BOHR PHENOMENON FOR CERTAIN CLOSE-TO-CONVEXANALYTIC FUNCTIONS
VASUDEVARAO ALLU AND HIMADRI HALDER
Abstract.
We say that a class B of analytic functions f of the form f ( z ) = P ∞ n =0 a n z n in the unit disk D := { z ∈ C : | z | < } satisfies a Bohr phenomenon iffor the largest radius R f < , the following inequality ∞ X n =1 | a n z n | ≤ d ( f (0) , ∂f ( D )) holds for | z | = r ≤ R f and for all functions f ∈ B . The largest radius R f iscalled Bohr radius for the class B . In this article, we obtain Bohr radius forcertain subclasses of close-to-convex analytic functions. We establish the Bohrphenomenon for certain analytic classes S ∗ c ( φ ) , C c ( φ ) , C ∗ s ( φ ) , K s ( φ ) . Using Bohrphenomenon for subordination classes [12, Lemma 1], we obtain some radius R f such that Bohr phenomenon for these classes holds for | z | = r ≤ R f . Generally, inthis case R f need not be sharp, but we show that under some additional conditionson φ , the radius R f becomes sharp bound. As a consequence of these results, weobtain several interesting corollaries on Bohr phenomenon for the aforesaid classes. Introduction and Preliminaries
Let f be an analytic function in the unit disk D := { z ∈ C : | z | < } with thefollowing power series representation(1.1) f ( z ) = ∞ X n =0 a n z n . Then the majorant series M f ( r ) associated with f given by (1.1), is defined by M f ( r ) := P ∞ n =0 | a n | r n for | z | = r < . The classical result of H. Bohr [14], which inthe sharp form has been independently proved by Weiner, Riesz and Schur reads asfollows: Theorem A.
Let f be analytic in D of the form (1.1) and | f ( z ) | < for all z ∈ D .Then the associated majorant series (1.2) M f ( r ) = ∞ X n =0 | a n | r n ≤ for | z | = r ≤ / and the constant / , referred to as the Bohr radius, cannot be improved. In the recent years, studying the Bohr radus has become an interesting problemin various directions in functions of one and several complex variables. The notionof Bohr radius has been extended to several complex variables, to planar harmonicmappings, to polynomials, to solutions of elliptic partial differential equations, andto more abstract settings. For more information and intriguing aspects about Bohr
File: Himadri-Vasu-P3-12-september-03-36-a-m.tex, printed: 2020-9-15, 1.34
Mathematics Subject Classification.
Primary 30C45, 30C50, 30C80.
Key words and phrases.
Starlike, convex, close-to-convex, quasi-convex functions; conjugatepoints, symmetric points; subordination, majorant series; Bohr radius. radius and Bohr inequality as stated above, we suggest the reader to glance throughthe articles [5, 9, 11, 13] and the references therein.The inequality (1.2) can also be written in the following form(1.3) ∞ X n =1 | a n z n | ≤ − | a | = d ( f (0) , ∂f ( D )) for | z | = r ≤ / , where d is the Euclidean distance. It is worth noting that theexistence of the radius / in (1.3) is independent of the coefficients of the powerseries (1.1). Analytic functions of the form (1.1) with modulus less than satisfyingthe inequality (1.3), are sometimes said to satisfy the classical Bohr phenomenon.Therefore we conclude that Bohr phenomenon occurs in the class of analytic self-maps of the unit disk D . The notion of Bohr phenomenon has been extended to theclass of analytic functions from D into a given domain D ⊆ C . Let G be the class ofanalytic functions of the form (1.1) which map D into a given domain D such that f ( D ) ⊆ D . Suppose there exists the largest radius r D > such that(1.4) ∞ X n =1 | a n z n | ≤ d ( f (0) , ∂f ( D )) in | z | ≤ r D for all functions f ∈ G . In this case, we say that G satisfies the Bohr phenomenon.It has been proved [6] that the largest radius r D for convex domain D coincides withthe classical Bohr radius / while Abu-Muhanna [1] has obtained r D = 3 − √ for any proper simply connected domain D . For more intriguing aspects of Bohrphenomenon, we refer the reader to the articles [2, 3, 7, 8]. The Bohr phenomenonfor certain subclasses of harmonic mappings has also been extensively studied byseveral authors [4, 10, 18].Let A denote the class of normalized analytic functions in D of the form(1.5) f ( z ) = z + ∞ X n =2 a n z n and S be its standard subclass made up of normalized univalent ( i.e. one-to-one)functions in D . An analytic function f in D is said to be subordinate to an analyticfunction g in D , denoted by f ≺ g (sometimes written as f ( z ) ≺ g ( z ) ), if f ( z ) = g ( ω ( z )) for z ∈ D , where ω : D → D is an analytic function such that ω (0) = 0 .In particular, when g is univalent in D , then f ≺ g if, and only if, f (0) = g (0) and f ( D ) ⊆ g ( D ) . Let φ : D → C be Ma-Minda function which is analytic and univalentin D such that φ ( D ) has positive real part, symmetric with respect to the real axis,starlike with respect to φ (0) = 1 and φ ′ (0) > . Such Ma-Minda functions have theseries representation of the form φ ( z ) = 1 + P ∞ n =1 B n z n ( B > . For such φ , Ma-Minda [23] have considered the classes S ∗ ( φ ) and C ( φ ) , called Ma-Minda type starlikeand Ma-Minda type convex classes associated with φ respectively, where S ∗ ( φ ) and C ( φ ) are the subclasses of functions in S such that zf ′ ( z ) /f ( z ) ≺ φ ( z ) and zf ′′ ( z ) /f ′ ( z ) ≺ φ ( z ) respectively. Clearly, f ∈ C ( φ ) if, and only if, zf ′ ∈ S ∗ ( φ ) . It isimportant to note that for every such φ described as above, S ∗ ( φ ) ( C ( φ ) respectively ) always a subclass of the well-known starlike class S ∗ ( convex class C respectively ) bytaking φ ( z ) = (1 + z ) / (1 − z ) . For more intriguing aspects and geometric propertiesof starlike and convex functions, we refer the book [28]. For various φ , the classes S ∗ ( φ ) and C ( φ ) yield various important subclasses of starlike and convex functions,respectively. When φ ( z ) = (1 + (1 − α )) / (1 − z ) , we obtain the classes S ∗ ( α ) and C ( α ) . By taking φ ( z ) = (1 + Az ) / (1 + Bz ) , S ∗ ( φ ) and C ( φ ) reduce to the ohr phenomenon for certain close-to-convex analytic functions 3 Janowski starlike class S ∗ [ A, B ] and Janowski convex class C [ A, B ] respectively. Bytaking φ ( z ) = ((1 + z ) / (1 − z )) α for < α ≤ , we obtain the classes of stronglyconvex and strongly starlike functions of order α . By choosing φ ( z ) = (1 + sz ) with < s ≤ / √ , the class S ∗ ( φ ) reduces to ST L ( s ) := S ∗ (cid:0) (1 + sz ) (cid:1) . Masih andKanas [24] have considered the class ST L ( s ) . Khatter et al. [19] have introducedthe class S ∗ α,e := S ∗ ( α + (1 − α ) e z ) for ≤ α < .The extremal functions k and h respectively for the classes C ( φ ) and S ∗ ( φ ) asfollows:(1.6) zk ′′ ( z ) k ′ ( z ) = φ ( z ) and zh ′ ( z ) h ( z ) = φ ( z ) with the normalizations k (0) = k ′ (0) − and h (0) = h ′ (0) − . Thefunctions k and h belong to the classes C ( φ ) and S ∗ ( φ ) and they play the role ofKoebe functions in the respective classes. Ma and Minda [23] have obtained thefollowing subordination result and growth estimates for the classes S ∗ ( φ ) and C ( φ ) . Lemma 1.7. [23]
Let f ∈ S ∗ ( φ ) . Then zf ′ ( z ) /f ( z ) ≺ zh ′ ( z ) /h ( z ) and f ( z ) /z ≺ h ( z ) /z . Lemma 1.8. [23]
Let f ∈ C ( φ ) . Then zf ′′ ( z ) /f ′ ( z ) ≺ zk ′′ ( z ) /k ′ ( z ) and f ′ ( z ) ≺ k ′ ( z ) . Ma-Minda functions φ have been considered with the condition φ ′ (0) > . Mo-tivated by this, recently, Kumar and Banga [22] have introduced the function Φ ,called non-Ma-Minda function, with the condition Φ ′ (0) < and the other condi-tions on Φ are same as that of φ . Note that Φ can obtained from φ by a rotation,namely, z by − z . By going a similar manner as the definition of S ∗ ( φ ) and C ( φ ) (see[23]), Kumar and Banga have considered the classes S ∗ (Φ) and C (Φ) and studiedthe growth estimates and other basic properties of these classes.A function f ∈ A is said to be close-to-convex if there exists g ∈ S ∗ such that Re ( zf ′ ( z ) /g ( z )) > for z ∈ D . Let K denote the class of close-to-convex functionsin D . In 1959, Sakaguchi [21] introduced the subclass S ∗ s of functions starlike withrespect to symmetric points, which consists of functions f ∈ S satisfying the con-dition Re ( zf ′ ( z ) / ( f ( z ) − f ( − z ))) > for z ∈ D . Motivated by S ∗ s , Wang et.al. [30] have considered the class C s . More precisely, a function f ∈ C s if f satisfies theinequality Re (cid:0) ( zf ′ ( z )) ′ / (cid:0) ( f ( z ) − f ( − z )) ′ (cid:1)(cid:1) > in D . A function f ∈ A is starlikewith respect to conjugate points and convex with respect to conjugate points in D respectively if f satisfies the conditions Re zf ′ ( z ) f ( z ) + f (¯ z ) ! > and Re ( zf ′ ( z )) ′ (cid:16) f ( z ) + f (¯ z ) (cid:17) ′ > for z ∈ D respectively. A function f ∈ A is starlike with respect to symmetric conjugate pointsin D if it satisfies the inequality Re zf ′ ( z ) f ( z ) − f ( − ¯ z ) ! > , z ∈ D . In more general, Ravichandran [26] has defined the classes S ∗ s ( φ ) and C s ( φ ) . Definition 1.1. [26] A function f ∈ A is in the class S ∗ s ( φ ) if zf ′ ( z ) f ( z ) − f ( − z ) ≺ φ ( z ) , z ∈ D Vasudevarao Allu and Himadri Halder and is in the class C s ( φ ) if zf ′ ( z )) ′ f ′ ( z ) + f ′ ( − z ) ≺ φ ( z ) , z ∈ D . Similarly, let S ∗ c ( φ ) and S ∗ sc ( φ ) be the corresponding classes of starlike functionswith respect to conjugate points and symmetric conjugate points respectively. Let C c ( φ ) and C sc ( φ ) be the corresponding classes of convex functions with respect toconjugate points and symmetric conjugate points respectively The following lemmasare required to prove our main results. Lemma 1.9. [26]
Let min | z | = r | φ ( z ) | = φ ( − r ) , max | z | = r | φ ( z ) | = φ ( r ) , | z | = r . If f ∈ C s ( φ ) , then r r Z φ ( − r )( k ′ ( − r )) / dr ≤ | f ′ ( z ) | ≤ r r Z φ ( r )( k ′ ( r )) / dr. From [30, Theorem 9], for f ∈ C s ( φ ) , we have(1.10) Z r s s Z φ ( − t )( k ′ ( − t )) / dt ds ≤ | f ( z ) | ≤ Z r s s Z φ ( t )( k ′ ( t )) / dt ds and the results are sharp for the following function(1.11) f ( z ) = Z z ξ ξ Z φ ( − η )( k ′ ( − η )) / dη dξ ∈ C s ( φ ) , since it belongs to the class C ( φ ) and having real coefficients. Lemma 1.12. [17]
Let f ( z ) = z + a l +1 z l +1 + · · · ∈ C ( φ ) , then we have ( k ′ ( − r l )) /l ≤ | f ′ ( z ) | ≤ ( k ′ ( r l )) /l . The bounds are sharp for some suitable rotations of the function K l which is definedby K l ( z ) = z Z ( k ′ ( ξ l )) /l dξ, z ∈ D , where k is defined in (1.6) . In particular for l = 2 we can obtain the bounds of | f ′ ( z ) | for odd convex functions.From Lemma 1.12, the following can be easily obtained for l = 2 r Z ( k ′ ( − t )) / dt ≤ | f ( z ) | ≤ r Z ( k ′ ( t )) / dt. The result is sharp for the function K = K is defined by K ( z ) := R z ( k ′ ( ξ )) / dξ .It is easy to see that K is odd convex function which belongs to C ( φ ) . Similary, wenote that the function H is defined by H ( z ) := ( h ( z )) / is a Koebe type functionfor odd starlike class in S ∗ ( φ ) and satisfies the relation(1.13) zK ′ ( z ) = H ( z ) . Lemma 1.14. [26]
Let min | z | = r | φ ( z ) | = φ ( − r ) , max | z | = r | φ ( z ) | = φ ( r ) , | z | = r . If f ∈ S ∗ c ( φ ) , then (i) h ′ ( − r ) ≤ | f ′ ( z ) | ≤ h ′ ( r ) ohr phenomenon for certain close-to-convex analytic functions 5 (ii) − h ( − r ) ≤ | f ( z ) | ≤ h ( r ) (iii) f ( D ) ⊇ { w : | w | ≤ − h ( − } .The results are sharp. Lemma 1.15. [26]
Let min | z | = r | φ ( z ) | = φ ( − r ) , max | z | = r | φ ( z ) | = φ ( r ) , | z | = r . If f ∈ C c ( φ ) , then (i) k ′ ( − r ) ≤ | f ′ ( z ) | ≤ k ′ ( r ) (ii) − k ( − r ) ≤ | f ( z ) | ≤ k ( r ) (iii) f ( D ) ⊇ { w : | w | ≤ − k ( − } .The results are sharp. Motivated by the class S ∗ s , Gao and Zhou [16] have studied the class K s of close-to-convex univalent functions, where K s is the class of functions f ∈ S satisfyingthe condition Re (cid:18) z f ′ ( z ) g ( z ) g ( − z ) (cid:19) < , z ∈ D . A more general class K s ( φ ) has been studied extensively by Cho et.al. [15] andWang et.al. [29]. For the brevity, we write the definition. Definition 1.2. [29] For a function φ with positive real part, the class K s ( φ ) consistsof functions f ∈ A satisfying − z f ′ ( z ) g ( z ) g ( − z ) ≺ φ ( z ) in D for some function g ∈ S ∗ (1 / .In particular, for φ ( z ) = (1 + (1 − γ ) z ) / (1 − z ) with ≤ γ < , the class K s ( φ ) reduces to K s ( γ ) which has recently been investigated by Kowalczyk and Les-Bomba[20]. When γ = 0 , we can obtain K s , the subclass of close-to-convex functions whichhas been defined by Gao and Zhou [16]. When φ ( z ) = (1 + βz ) / (1 − αβz ) , where ≤ α ≤ and < β ≤ , the class K s ( φ ) reduces to K s ( α, β ) defined in [29]. Nowlet q ( z ) = P ∞ n =1 q n z n be analytic in D . Then for fixed f ∈ K s ( φ ) , we define(1.16) S K f ( φ ) := ( q ( z ) = ∞ X n =1 q n z n : q ≺ f ) . The distortion and growth theorems for the class K s ( φ ) have been obtained in [15].Let φ be a Ma-Minda function. Lemma 1.17. [15]
Let min | z | = r | φ ( z ) | = φ ( − r ) , max | z | = r | φ ( z ) | = φ ( r ) , | z | = r . If f ∈ K s ( φ ) , then the following sharp inequalities hold: (i) φ ( − r )1 + r ≤ | f ′ ( z ) | ≤ φ ( r )1 − r ( | z | = r < (ii) r Z φ ( − t )1 + t dt ≤ | f ( z ) | ≤ r Z φ ( t )1 − t dt ( | z | = r < . Vasudevarao Allu and Himadri Halder
Let f and g be two analytic functions in D such that g ≺ f . Let(1.18) g ( z ) = ∞ X n =0 b n z n . In 2018, Bhowmik and Das [12] proved the following interesting result for subordi-nation classes.
Lemma 1.19. [12]
Let f and g be analytic in D with Taylor expansions (1.1) and (1.18) respectively and g ≺ f , then (1.20) ∞ X n =0 | b n | r n ≤ ∞ X n =0 | a n | r n for z | = r ≤ / . In general, one obtains the Bohr radius for certain classes of analytic functionsin D , when the sharp coefficient bounds for this class are known. But the sharpcoefficient bounds for most of the Ma-Minda subclasses are not yet known. UsingLemma 1.19, Allu and Halder [11] recently have obtained Bohr radius for certainclasses of Ma-Minda starlike and convex functions. In this article, we consider cer-tain classes of close-to-convex functions associated with Ma-Minda functions e.g. S ∗ c ( φ ) , C c ( φ ) , K s ( φ ) and C s ( φ ) . The sharp coefficient bounds of these classes arenot yet known. Hence, we encounter the problem to find the best possible lowerbound of the radius so that Bohr phenomenon holds for these classes. As a conse-quence, we also establish the Bohr phenomenon for several important subclasses forparticular choices of φ . 2. Main Results
Before going to state our main results we prove an preliminary result which isrequired to prove some of our results.
Lemma 2.1. (i)
Let f and g be analytic in D with series representation f ( z ) = P ∞ n =1 a n z n and (1.18) respectively such that f ( z ) = R z g ( ξ ) dξ for z ∈ D ,where integration is taken along a linear segment joining to z ∈ D . Then M f ( r ) = Z r M g ( t ) dt for | z | = r < . Here M f ( r ) and M g ( r ) are respectively the majorant series associated with f and g respectively. (ii) Let f and g be analytic in D with Taylor expansions (1.1) and (1.18) re-spectively and g ≺ f , then M G ( r ) ≤ M F ( r ) for | z | = r ≤ / , where G ( z ) = R z g ( ξ ) dξ and F ( z ) = R z f ( ξ ) dξ for z ∈ D . Let min | z | = r | φ ( z ) | = φ ( − r ) and max | z | = r | φ ( z ) | = φ ( r ) , | z | = r . We assume thesenotations throught this paper. Here φ is the Ma-Minda function. Theorem 2.2.
Let f ∈ K s ( φ ) be of the form (1.5) . Then (2.3) | z | + ∞ X n =2 | a n || z | n ≤ d ( f (0) , ∂f ( D )) for | z | = r ≤ R f , where R f = min { / , r f } and r f is the smallest positive root of R ( r ) = L (1) in (0 , . Here R ( r ) := R r ( M φ ( t )) / (1 − t ) dt , L ( r ) := R r ( φ ( − t )) / (1+ t ) dt and M φ is the associated majorant series of φ . ohr phenomenon for certain close-to-convex analytic functions 7 Remark 2.1. (i) Assume that the coeficients of φ ( z ) = 1 + P ∞ n =1 B n z n in theTheorem 2.2 are all positive i.e. B n > for n ≥ . Then the majorant series M φ ( r ) = φ ( r ) , < r < and hence R ( r ) := R r ( φ ( t )) / (1 − t ) dt .(ii) (Bohr phenomenon for the corresponding class K s (Φ) associated with non-Ma-Minda functions) Let Φ be the corresponding non-Ma-Minda function of φ , which is actually a rotation by mere replacing z by − z . Therefore theimage of the unit disk D under the functions Φ and φ are identical. Thus weconclude that K s (Φ) = K s ( φ ) and the Bohr phenomenon (2.3) holds for theclass K s (Φ) for the same R f . Some applications:Lemma 2.4. ( Bohr phenomenon for the corresponding subordination class ) Let q ( z ) = P ∞ n =1 q n z n ∈ S K f ( φ ) as defined in (1.16) and f be of the form (1.5) . Then ∞ X n =1 | q n || z | n ≤ d ( f (0) , ∂f ( D )) for | z | = r ≤ R f , where R f is defined as in Theorem 2.2. For φ ( z ) = (1 + (1 − γ ) z ) / (1 − z ) , the class K s ( φ ) reduces to K s ( γ ) . In particular,for γ = 0 , K s ( φ ) reduces to K s . Corollary 2.5. (i) ( Bohr phenomenon for the class K s ( γ )) Any function f ∈ K s ( γ ) with ≤ γ < . satisfies the inequality (2.3) for | z | = r ≤ r f , where r f is the root of (2.6) γ ln (cid:18) r − r (cid:19) + (1 − γ ) r − r = 1 − γ ln γπ in (0 , / . (ii) Each function f ∈ K s satisfies the Bohr inequality (2.3) for | z | = r ≤ r f ,where r f = ln / (2 + ln ≈ . . For φ ( z ) = (1 + βz ) / (1 − αβz ) , where ≤ α ≤ and < β ≤ , the class K s ( φ ) reduces to K s ( α, β ) . In particular, for α = β = 1 , K s ( α, β ) coincides with the class K s . Corollary 2.7.
The class K s ( α, β ) satisfies the Bohr phenomenon (2.3) for | z | = r ≤ R f = min { / , r f } , where r f is the smallest root of (2.8) r Z βt (1 − αβt )(1 − t ) dt = Z − βt (1 + αβt )(1 + t ) dt in (0 , . Theorem 2.9.
Let f ∈ S ∗ c ( φ ) be of the form (1.5) . Then (2.10) | z | + ∞ X n =2 | a n || z | n ≤ d ( f (0) , ∂f ( D )) for | z | = r ≤ min { / , r f } and r f is the smallest positive root of P ( r ) + h ( −
1) = 0 in (0 , , where P ( r ) := R r (( M h ( t ) M φ ( t )) /t ) dt . Here M h ( t ) and M φ ( t ) are themajorant series of h and φ respectively. Remark 2.2. (i) ( Bohr radius for S ∗ c ( φ ) when φ has positive coefficients ) Let φ ( z ) = 1 + P ∞ n =1 B n z n . It is worth to point out that if we impose an Vasudevarao Allu and Himadri Halder additional condition on φ that the coefficients B n ’s are positive, then themajorant series M φ ( r ) = φ ( r ) . From the definition of h in (1.6), we obtain(2.11) h ( z ) = z exp z Z φ ( t ) − t dt = z exp ∞ X n =1 B n n z n ! . From (2.11), it is easy to see that M h ( r ) = h ( r ) and P ( r ) = Z r (( h ( t ) φ ( t )) /t ) dt = h ( r ) . Then each f ∈ S ∗ c ( φ ) satisfies the inequality (2.10) for | z | ≤ min { / , r f } ,where r f is the root of the equation h ( r ) + h ( −
1) = 0 . In particular, when r f ≤ / , the radius r f is the best possible for the function f = h ∈ S ∗ c ( φ ) ,since it has real coefficients and belongs to S ∗ ( φ ) . Indeed, for | z | = r f , M h ( r f ) = h r f = − h ( −
1) = d ( h (0) , ∂h ( D )) , which shows that r f is the bestpossible.(ii) (Bohr phenomenon for corresponding class S ∗ c (Φ) associated with non-Ma-Minda function) Let Φ be the corresponding non-Ma-Minda function of φ .Since Φ is actually obtained from φ by a rotation z by − z , the image of theunit disk D under the functions Φ and φ are identical. Thus we concludethat S ∗ c (Φ) = S ∗ c ( φ ) and the Bohr radius for the class S ∗ c (Φ) is same as thatof S ∗ c ( φ ) .Let S ∗ cf ( φ ) denote the class of analytic functions g which are subordinate to afixed function f ∈ S ∗ c ( φ ) . Lemma 2.12. (cid:0)
Bohr phenomenon for the corresponding subordination class S ∗ cf ( φ ) (cid:1) Let g ∈ S ∗ cf ( φ ) be of the form g ( z ) = P ∞ n =1 g n z n . Then (2.13) ∞ X n =1 | g n || z | n ≤ d ( f (0) , ∂f ( D )) for | z | = r ≤ min { / , r f } , where r f is given as in Theorem 2.9. Similar results on Bohr phenomenon for the class S ∗ c ( φ ) hold for the class S ∗ cf ( φ ) .In view of the Remark 2.2 and Lemma 2.12, we obtain the following interestingcorollaries. Let φ ( z ) = (1 + sz ) with . < s ≤ / √ , then S ∗ c ( φ ) reduces tothe class S ∗ c ((1 + sz ) ) . Corollary 2.14.
The class S ∗ c ((1 + sz ) ) (cid:0) and S ∗ cf ((1 + sz ) ) (cid:1) satisfies the Bohrinequality (2.10) for | z | = r ≤ r f , where < r f < / and r f is the root of theequation (2.15) r exp (cid:18) s (cid:18) r + sr (cid:19)(cid:19) = exp (cid:16) s (cid:16) − s (cid:17)(cid:17) . The radius r f is the best posible. From Table 1, it is easy to see that r f > / when s < . and hence Bohrphenomenon holds for r ≤ / and r f < / when . < s ≤ / √ . Thereforethe radius r f is the best possible. Corollary 2.16.
For φ ( z ) = α + (1 − α ) e z with ≤ α < . , the class S ∗ c ( φ ) satisfies the Bohr phenomenon (2.10) for | z | = r ≤ r f , where < r f < / . Therdius r f is the best possible. ohr phenomenon for certain close-to-convex analytic functions 9 s r f . . .
15 0 . . . .
25 0 . . . .
35 0 . . . s r f .
45 0 . . . .
55 0 . . . .
65 0 . . . / √ . Table 1.
The radius r f for different values of sα h (1 / h ( − Sign of D (0) Sign of D (1 / . . . − +0 .
01 0 . . − +0 .
02 0 . . − +0 .
03 0 . . − +0 .
04 0 . . − +0 .
05 0 . . − +0 .
06 0 . . − − .
07 0 . . − − Table 2.
Existance of the sharp radius r f in (0 , / for differentvalues of α in [0 , . From Table 2, it is clear that r f lies in (0 , / when ≤ α < . andhence r f is the best posiible. On the other hand r f > / for α > . and thecorresponding Bohr phenomenon holds for r ≤ / . Corollary 2.17.
Let φ ( z ) = ((1 + z ) / (1 − z )) α with < α ≤ . Also assume h (1 / > − h ( − , where h ( r ) = r exp r Z (cid:0) t − t (cid:1) α − t dt and − h ( −
1) = exp − Z (cid:0) t − t (cid:1) α − t dt . Then the class S ∗ c ( φ ) satisfies the Bohr phenomenon (2.10) for | z | = r ≤ r f , where r f is the smallest root of the equation D ( r ) := h ( r ) + h ( −
1) = 0 . From the Table 3, it is evident that for different values of α , the constant r f some-times does not lie in (0 , / . However, when r f lies in (0 , / , the corresponding r f is the best possible and the Bohr phenomenon for the class S ∗ c ( φ ) holds for r ≤ r f . Corollary 2.18.
Let φ ( z ) = (1 + (1 − γ ) z ) / (1 − z ) with ≤ γ < / . Then each f ∈ S ∗ c ((1 + (1 − γ ) z ) / (1 − z )) satisfies the inequality (2.10) for | z | = r ≤ r f ,where < r f < / and r f is the root of (2.19) r + 2 r / (2(1 − γ )) − . The radius r f is the best possible. α h (1 / − h ( − Sign of D (0) Sign of D (1 / . . . − − . . . − − .
45 0 . . − +0 . . . − +0 . . . − +0 . . . − +0 . . . − +0 . . . − + Table 3.
Existance of the sharp radius r f in (0 , / for differentvalues of α Corollary 2.20. If φ ( z ) = (1 + Az ) / (1 + Bz ) with − ≤ B < A ≤ , then (i) when B = 0 , every function f ∈ S ∗ c ((1 + Az ) / (1 + Bz )) satisfies the inequal-ity (2.10) for | z | = r ≤ r f , where < r f < / and r f is the unique rootof (2.21) re Ar = e − A , provided A ≥ (3 / ln . The radius r f is the best possible. (ii) When B = 0 , every function f ∈ S ∗ c ((1 + Az ) / (1 + Bz )) satisfies the in-equality (2.10) for | z | = r ≤ r f , where < r f < / and r f is the uniqueroot of (2.22) r (1 + Br ) A − BB = (1 − B ) A − BB , provided (1 /
3) (1 + B/ ( A − B ) /B ≥ (1 − B ) ( A − B ) /B . The radius r f is the bestpossible. ( A = 1) B r f − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . A = 1 / B r f − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . Table 4.
The radius r f for different values of B when A = 1 and A = 1 / From the Table , we see that for different values of A and B , sometimes theradius r f < / . and in that case r f is the best possible. When r f > / ,Bohr phenomenon for class S ∗ c ((1 + Az ) / (1 + Bz )) holds for r ≤ / . Theorem 2.23.
Let f ∈ C c ( φ ) be of the form (1.5) . Then (2.24) | z | + ∞ X n =2 | a n || z | n ≤ d ( f (0) , ∂f ( D )) ohr phenomenon for certain close-to-convex analytic functions 11 for | z | = r ≤ min { / , r f } and r f is the smallest positive root of T ( r ) = − k ( − in (0 , and T ( r ) := r Z s s Z M k ′ ( t ) M φ ( t ) dt ds. Here M k ′ ( t ) and M φ ( t ) are respectively the majorant series of k ′ and φ respectively. Theorem 2.25.
Let f ∈ C s ( φ ) be of the form (1.5) . Then (2.26) | z | + ∞ X n =2 | a n || z | n ≤ d ( f (0) , ∂f ( D )) for | z | = r ≤ min { / , r f } and r f is the smallest positive root of R s ( r ) = L s (1) in (0 , , where R s ( r ) := r Z s s Z M K ′ ( t ) M φ ( t ) dt ds and L s ( r ) := r Z s s Z (cid:0) k ′ ( − t ) (cid:1) / φ ( − t ) dt ds and K ′ ( r ) = ( k ′ ( t )) / . Remark 2.3. (i) Let Φ be the corresponding non-Ma-Minda class with respectto φ . Then the Bohr radius for the class C s (Φ) is same as that of C s ( φ ) .(ii) Let S ∗ sf ( φ ) be the class of analytic functions g of the form g ( z ) = P ∞ n =1 g n z n in D subordinate to a fixed function f ∈ C s (Φ) , then ∞ X n =1 | g n || z | n ≤ d ( f (0) , ∂f ( D )) for | z | = r ≤ min { / , r f } and r f is given as in Theorem 2.25.3. Proof of the main results
Proof of Lemma 2.1. (i) In view of the relation f ( z ) = R z g ( ξ ) dξ , we obtain ∞ X n =1 a n z n = ∞ X n =1 b n − n z n . Therefore M f ( r ) = ∞ X n =1 | b n − | n r n = r Z ∞ X n =0 | b n | t n dt = Z r M g ( t ) dt for r < . (ii) From Lemma 1.19, we have M g ( r ) ≤ M f ( r ) for r ≤ / and integrating thiswe obtain r Z M g ( t ) dt ≤ r Z M f ( t ) dt for r ≤ / . Hence from the first part of this Lemma, we obtain M G ( r ) = r Z M g ( t ) dt ≤ r Z M f ( t ) dt = M F ( r ) for r ≤ / . (cid:3) Proof of Theorem 2.2.
Let f ∈ K s ( φ ) , then from Lemma 1.17, the Euclideandistance between f (0) and the boundary of f ( D ) is(3.1) d ( f (0) , ∂f ( D )) = lim inf | z |→ | f ( z ) − f (0) | ≥ Z φ ( − t )1 + t dt. By the subordination principle, there exists an analytic function ω : D → D with ω (0) = 0 such that(3.2) − z f ′ ( z ) g ( z ) g ( − z ) = φ ( ω ( z )) . Let G ( z ) := − g ( z ) g ( − z ) /z . Clearly, G is an odd starlike function in D . Let G ( z ) = z + P ∞ n =2 g n − z n − . It is well-known that | g n − | ≤ for n ≥ . Therefore(3.3) M G ( r ) ≤ r + ∞ X n =2 r n − = r − r , < r < . From (3.2), we have zf ′ ( z ) = G ( z ) φ ( ω ( z )) , which immediately follows that(3.4) f ( z ) = z Z G ( ξ ) φ ( ω ( ξ )) ξ dξ. It is known that for two analytic functions f and g in D , M fg ( r ) ≤ M f ( r ) M g ( r ) ,where M f ( r ) , M g ( r ) and M fg ( r ) are associated majorant series of f , g and theproduct f g respectively. Therefore M G ( φ ◦ ω ) ( r ) ≤ M G ( r ) M φ ◦ ω ( r ) . Since φ ◦ ω ≺ φ ,by Lemma 1.19, we have(3.5) M φ ◦ ω ( r ) ≤ M φ ( r ) for | z | = r ≤ / . In view of Lemma 2.1 and (3.3), (3.4) and (3.5), we obtain(3.6) M f ( r ) ≤ r Z M G ( t ) M φ ◦ ω ( t ) t dt ≤ r Z M φ ( t )1 − t dt = R ( r ) for | z | = r ≤ / . We note that R ( r ) ≤ L (1) whenever r ≤ r f , where r f is thesmallest positive root of R ( r ) = L (1) in (0 , . Let H ( r ) = R ( r ) − L (1) then H ( r ) is continuous function in [0 , . Since R (1) > L (1) and M φ ( t ) ≥ | φ ( t ) | , it followsthat H (0) = L (1) = − Z φ ( − t )1 + t dt < and H (1) = R (1) − L (1) = Z M φ ( t )1 − t dt − Z φ ( − t )1 + t dt > . Therefore H has a root in (0 , . Let r f be the smallest root of H in (0 , . Then R ( r ) ≤ L (1) for r ≤ r f . From (3.1) and (3.6), we obtain M f ( r ) ≤ Z φ ( − t )1 + t dt ≤ d ( f (0) , ∂f ( D )) ohr phenomenon for certain close-to-convex analytic functions 13 for | z | = r ≤ min { / , r f } = R f . (cid:3) Proof of Lemma 2.4.
From the definition of S K f ( φ ) , we have q ≺ f . In view ofLemma 1.19, we obtain M q ( r ) ≤ M f ( r ) for | z | = r ≤ / . Hence from (2.3), we get P ∞ n =1 | q n || z | n ≤ d ( f (0) , ∂f ( D )) for | z | = r ≤ min { / , r f } . (cid:3) Proof of Corollary 2.5. (i) Let f ∈ K s ( γ ) . Then a simple computation shows that R ( r ) = γ ln (cid:18) r − r (cid:19) + (1 − γ ) r − r and L ( r ) = (1 − γ ) ln (cid:18) r √ r (cid:19) + γ arctan r. Clearly, L (1) = ((1 − γ ) / ln γπ/ and H ( r ) := R ( r ) − L (1) . Then H is continuous in [0 , . A simple computation shows that H (0) < and H (1 / > if ≤ γ < . . Therefore, H has a root in (0 , / and choose the smallest root to be r f in (0 , / . Thus the inequality (2.3)holds for | z | = r ≤ r f .(ii) Putting γ = 0 in (2.6), we obtain r f = ln / (2 + ln . (cid:3) Proof of Theorem 2.7.
It is easy to see that the coefficients of the power seriesof φ ( z ) = (1 + βz ) / (1 − αβz ) are positive, where ≤ α ≤ and < β ≤ . In viewof Remark 2.1 (i), we obtain M φ ( r ) = φ ( r ) and R ( r ) = r Z βt (1 − αβt )(1 − t ) dt. Therefore, from Theorem 2.2, r f is the root of r Z βt (1 − αβt )(1 − t ) dt = Z − βt (1 + αβt )(1 + t ) dt. Thus, the class K s ( α, β ) satisfies the Bohr phenomenon (2.3) for | z | = r ≤ R f =min { / , r f } . (cid:3) Proof of Theorem 2.9.
Let f ∈ S ∗ c ( φ ) , then by using Lemma 1.14 we obtain thefollowing Euclidean distance between f (0) and the boundary of f ( D ) as(3.7) d ( f (0) , ∂f ( D )) = lim inf | z |→ | f ( z ) − f (0) | ≥ − h ( − . Since f ∈ S ∗ c ( φ ) and φ is starlike and symmetric with respect to real-axis, it followsthat g ( z ) := ( f ( z ) + f (¯ z )) / belongs to S ∗ ( φ ) . Since g ∈ S ∗ ( φ ) , from Lemma 1.7,we have g ( z ) /z ≺ h ( z ) /z . Therefore from Lemma 1.19, we obtain(3.8) M g ( r ) ≤ M h ( r ) for | z | = r ≤ / . From the definition of S ∗ c ( φ ) , we have(3.9) zf ′ ( z ) = g ( z ) φ ( ω ( z )) , where ω : D → D is analytic with ω (0) = 0 . Since φ ◦ ω ≺ ω , from Lemma 1.19 weobtain(3.10) M φ ◦ ω ( r ) ≤ M φ ( r ) for | z | = r ≤ / . A simplification of (3.9) gives(3.11) f ( z ) = z Z g ( ξ ) φ ( ω ( ξ )) ξ dξ. Now, by making use of Lemma 2.1 as well as (3.8) and (3.10) in (3.11), we obtain | z | + ∞ X n =2 | a n || z | n = M f ( r ) (3.12) ≤ r Z M g ( t ) M φ ◦ ω ( t ) t dt ≤ r Z M h ( t ) M φ ( t ) t dt = P ( r ) for | z | = r ≤ / . We note that P ( r ) ≤ − h ( − , whenever r ≤ r f , where r f isthe smallest positive root of P ( r ) = − h ( − in (0 , . Going by the similar line ofargument as in the proof of Theorem 2.2, the existance of the root r f is ensured bythe following inequalities M h ( t ) ≥ | h ( t ) | , M h (1) ≥ | h (1) | ≥ − h ( − and M h (0) < − h ( − . Thus, combining the inequalities (3.12) and (3.7) with the fact P ( r ) ≤ − h ( − for r ≤ r f , we conclude that | z | + ∞ X n =2 | a n || z | n ≤ d ( f (0) , ∂f ( D )) for | z | = r ≤ min { / , r f } . (cid:3) Proof of Lemma 2.12.
From the definition of S ∗ cf ( φ ) , we have g ≺ f . Then byLemma 1.19, we obtain M g ( r ) ≤ M f ( r ) for | z | = r ≤ / . Hence from (2.10), weobtain P ∞ n =1 | g n || z | n ≤ d ( f (0) , ∂f ( D )) for | z | = r ≤ min { / , r f } . (cid:3) Proof of Corollary 2.14.
Since the coefficients of φ ( z ) = (1 + sz ) with < s ≤ / √ are all positive, in view of Remark 2.2, we obtain P ( r ) = h ( r ) = r exp (cid:18) s (cid:18) r + sr (cid:19)(cid:19) . Let D ( r ) = h ( r )+ h ( − . Clearly D is continuous in [0 , . Observe that D (0) < and D (cid:18) (cid:19) = 13 exp (cid:18) s (cid:18) s + 1218 (cid:19)(cid:19) − exp (cid:16) s (cid:16) − s (cid:17)(cid:17) > , whenever . < s ≤ / √ . Therefore D has a real root in (0 , / and chooseit to be r f . Thus, from Remark 2.2, the radius r f is the best possible. (cid:3) ohr phenomenon for certain close-to-convex analytic functions 15 Proof of Corollary 2.16.
Let φ ( z ) = α + (1 − α ) e z . Then the coefficients ofthe Maclaurin series of φ ( z ) are positive for ≤ α < . Let D ( r ) = h ( r ) + h ( − ,where h ( r ) = r exp (1 − α ) r Z (cid:18) − e t t (cid:19) dt . It is easy to see that h (cid:18) (cid:19) = 13 exp (1 − α ) Z (cid:18) − e t t (cid:19) dt ≈
13 (1 . − α and h ( −
1) = − exp (1 − α ) − Z (cid:18) − e t t (cid:19) dt ≈ − (0 . − α . A simple computation shows that D (1 /
3) = h (1 / h ( − > if ≤ α < . .Clearly, D (0) = h ( − < . Therefore, D has a root in (0 , / and choose it tobe r f . In view of Remark 2.2, r f is the best possible. (cid:3) Proof of Corollary 2.17.
Let φ ( z ) = ((1 + z ) / (1 − z )) α with < α ≤ . From[4], it is guaranted that the coeffficients of the Maclaurin series of φ are positive. Itis easy to see that h ( r ) = r exp r Z (cid:0) t − t (cid:1) α − t dt . Then D ( r ) := h ( r ) + h ( − is continuous in [0 , and D (0) < and D (1 /
3) = h (1 /
3) + h ( − > . Thus D has a root in (0 , and choose it to be r f . Hence, inview of Remark 2.2, r f is the best possible. (cid:3) Proof of Corollary 2.18.
Let φ ( z ) = (1 + (1 − γ ) z ) / (1 − z ) . Then h ( z ) = z/ (1 − z ) − γ ) . It is easy to see that h (1 /
3) = 3 − γ ) − − γ ) and − h ( −
1) = 12 − γ ) . Further, h (1 / > − h ( − for ≤ γ ≤ / . Therefore (2.19) has a root in (0 , / and monotonocity of h ensures that this root is unique in (0 , / . Hence by theRemark 2.2, r f is the best possible for the class S ∗ c ((1 + (1 − γ ) z ) / (1 − z )) . (cid:3) Proof of Corollary 2.20.
When φ ( z ) = (1 + Az ) / (1 + Bz ) , then from (2.11) weobtain h ( z ) = z (1 + Bz ) A − BB , B = 0 ze Az , B = 0 . (i) When B = 0 , then h ( r ) = re Ar and − h ( −
1) = e − A . We note that h (1 / > − h ( − whenever (1 / e A/ > e − A . That is when A > (3 / ln . Therefore(2.21) has a root in (0 , / and choose r f be the smallest root in (0 , / .Hence r f is the best possible. (ii) If B = 0 , then h ( r ) = r (1 + Br ) ( A − B ) /B . It is easy to see that h (1 / > − h ( − when (1 /
3) (1 + B/ ( A − B ) /B ≥ (1 − B ) ( A − B ) /B . Therefore (2.22)has a root in (0 , / and choose r f to be the smallest root in (0 , / . Hence r f is the best possible. (cid:3) Proof of Theorem 2.23.
The proof of Theorem 2.23 follows from Theorem 2.9and the fact that zf ′ ∈ S ∗ c ( φ ) if, and only, if f ∈ C c ( φ ) . For the bravity we completethe proof. Let g ( z ) := ( f ( z ) + f (¯ z )) / . Since φ is starlike and symmetric withrespect to real axis, g ∈ C ( φ ) . From the definition of C c ( φ ) , we have(3.13) ( zf ′ ( z )) ′ = g ′ ( z ) φ ( ω ( z )) , where ω : D → D is analytic with ω (0) = 0 . A simple computation using (3.13)shows that(3.14) f ( z ) = z Z ξ ξ Z g ′ ( η ) φ ( ω ( η )) dη dξ. Since g ∈ C ( φ ) , in view of Lemma 1.8, we have g ′ ≺ k ′ and hence by Lemma 1.19,we obtain(3.15) M g ′ ( r ) ≤ M k ′ ( r ) for r ≤ / . In view of Lemma 2.1 and by using (3.14) and (3.15), we obtain(3.16) M f ( r ) ≤ r Z s s Z M k ′ ( t ) M φ ( t ) dt ds = T ( r ) for r ≤ / . From Lemma 1.15, the Euclidean distance between f (0) and the boundary of f ( D ) is(3.17) d ( f (0) , ∂f ( D )) = lim inf | z |→ | f ( z ) − f (0) | ≥ − k ( − . Clearly, T ( r ) ≤ − k ( − for r ≤ r f , where r f is the smallest positive root of T ( r ) = − k ( − in (0 , . Going by the similar lines of argument as in the proof of Theorem2.9, the existance of the root r f is ensured by the following inequalities M k ( r ) ≥ | k ( r ) | , M k (1) ≥ | k (1) | ≥ − k ( − and M k (0) < − k ( − . Therefore from (3.16) and (3.17), we obtain | z | + ∞ X n =0 | a n || z | n = M f ( r ) ≤ d ( f (0) , ∂f ( D )) for | z | = r ≤ min { / , r f } . (cid:3) Proof of Theorem 2.25.
Let f ∈ C s ( φ ) , then it is evident that the Euclideandistance between f (0) and the boundary of f ( D ) is(3.18) d ( f (0) , ∂f ( D )) = lim inf | z |→ | f ( z ) − f (0) | ≥ L s (1) . Since f ∈ C s ( φ ) and φ is starlike and symmetric with respect to the real axis, thenit follows that(3.19) g ( z ) := f ( z ) − f ( − z )2 = z + ∞ X n =1 a n +1 z n +1 ∈ C ( φ ) . ohr phenomenon for certain close-to-convex analytic functions 17 Here g is an odd convex analytic function. Note that the function K ( z ) = z R ( k ′ ( ξ )) / dξ defined in (1.13) is an odd function in C ( φ ) . By Lemma 1.8 we have g ′ ≺ K ′ . There-fore from Lemma 1.19, we obtain(3.20) M g ′ ( r ) ≤ M K ′ ( r ) for | z | = r ≤ / . From the definition of C s ( φ ) , we have(3.21) ( zf ′ ( z )) ′ = g ′ ( z ) φ ( ω ( z )) . A simplication of (3.21) gives(3.22) f ( z ) = z Z ξ ξ Z g ′ ( η ) φ ( ω ( η )) dη dξ. By making use of Lemmas 1.14 and 2.1 and in view of (3.20) and (3.22), we obtain | z | + ∞ X n =2 | a n || z | n = M f ( r ) ≤ r Z s s Z M g ′ ( t ) M φ ( t ) dt ds (3.23) ≤ r Z s s Z M K ′ ( t ) M φ ( t ) dt ds = R s ( r ) , for z | = r ≤ / . A simple observation shows that R s ( r ) ≤ L s (1) for r ≤ r f , where r f is the smallest root of R s ( r ) = L s (1) in (0 , . The existance of the root is ensuredby the following inequalities M K ′ ( t ) ≥ | K ′ ( t ) | , R s (1) ≥ L s (1) and R s (0) ≤ L s (1) as well as the inequality (1.10). Using (3.18) and (3.23) with the fact that R s ( r ) ≤ L s (1) for r ≤ r f , we obtain | z | + ∞ X n =2 | a n || z | n ≤ d ( f (0) , ∂f ( D )) for | z | = r ≤ min { / , r f } . This completes the proof. (cid:3)
Acknowledgement:
The first author thanks SERB-MATRICS and the secondauthor thanks CSIR for their support.
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Vasudevarao Allu, School of Basic Science, Indian Institute of TechnologyBhubaneswar, Bhubaneswar-752050, Odisha, India.
E-mail address : [email protected] Himadri Halder, School of Basic Science, Indian Institute of Technology Bhubaneswar,Bhubaneswar-752050, Odisha, India.
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