Starlikeness of Analytic Functions with Subordinate Ratios
aa r X i v : . [ m a t h . C V ] J a n Starlikeness of Analytic Functionswith Subordinate Ratios
Rosihan M. Ali, Kanika Sharma, and V. Ravichandran
Abstract.
Let h be a non-vanishing analytic function in the open unit disc with h (0) = 1. Consider the class consisting of normalized analytic functions f whose ratios f ( z ) /g ( z ), g ( z ) /zp ( z ), and p ( z ) are each subordinate to h for some analytic functions g and p . The radius of starlikeness is obtained for this class when h is chosen to beeither h ( z ) = √ z or h ( z ) = e z . Further G -radius is also obtained for each of thesetwo classes when G is a particular widely studied subclass of starlike functions. Theseinclude G consisting of the Janowski starlike functions, and functions which are parabolicstarlike.
1. Classes of Analytic Functions
Let A denote the class of normalized analytic functions f ( z ) = z + P ∞ k =2 a k z k inthe unit disc D = { z ∈ C : | z | < } . A prominent subclass of A is the class S ∗ consisting of functions f ∈ A such that f ( D ) is a starlike domain with respect to theorigin. Geometrically, this means the linear segment joining the origin to every otherpoint w ∈ f ( D ) lies entirely in f ( D ). Every starlike function in A is necessarily univalent.Since f ′ (0) does not vanish, every function f ∈ A is locally univalent at z = 0. Further,each function f ∈ A mirrors the identity mapping near the origin, and thus in particular,maps small circles | z | = r onto curves which bound starlike domains. If f ∈ A is alsorequired to be univalent in D , then it is known that f maps the disc | z | < r onto a domainstarlike with respect to the origin for every r ≤ r := tanh( π/
4) (see [ , Corollary, p. 98]).The constant r cannot be improved. Denoting by S the class of univalent functions f ∈ A , the number r = tanh( π/
4) is commonly referred to as the radius of starlikenessfor the class S .Another informative description of the class S is its radius of convexity. Here it isknown that every f ∈ S maps the disc | z | < r onto a convex domain for every r ≤ r :=2 − √ , Corollary, p. 44]. Thus the radius of convexity for S is r = 2 − √ G and M of A . The G -radius for the class M , denotedby R G ( M ), is the largest number R such that r − f ( rz ) ∈ G for every 0 < r ≤ R and f ∈ M . Thus, for example, an equivalent description of the radius of starlikeness for S is that the S ∗ -radius for the class S is R S ∗ ( S ) = tanh( π/ Mathematics Subject Classification.
Key words and phrases.
Starlike functions; subordination; radius of starlikeness.
In this paper, we seek to determine the radius of starlikeness, and certain other G -radius, for particular subclasses G of A . Several widely-studied subclasses of A havesimple geometric descriptions; these functions are often expressed as a ratio between twofunctions. Among the very early studies in this direction is the class of close-to-convexfunctions introduced by Kaplan [ ], and Reade’s class [ ] of close-to-starlike functions.Close-to-convex functions are necessarily univalent, but not so for close-to-starlike func-tions. Several works, for example those in [ , – , , ], have advanced studies inclasses of functions characterized by the ratio between functions f and g belonging togiven subclasses of A .In this paper, we examine two different subclasses of functions in A satisfying a certainsubordination link of ratios. Interestingly, these classes contain non-univalent functions.An analytic function f is subordinate to an analytic function g , written f ≺ g , if f ( z ) = g ( w ( z )) , z ∈ D , for some analytic self-map w in D with | w ( z ) | ≤ | z | . The function w is often referred toas a Schwarz function.Now let h be a non-vanishing analytic function in D with h (0) = 1. The classes treatedin this paper consist of functions f ∈ A whose ratios f ( z ) /g ( z ), g ( z ) /zp ( z ), and p ( z ), areeach subordinate to h for some analytic functions g and p : f ( z ) g ( z ) ≺ h ( z ) , g ( z ) zp ( z ) ≺ h ( z ) , p ( z ) ≺ h ( z ) . When p is the constant one function, then the class contains functions f ∈ A satisfyingthe subordination of ratios f ( z ) g ( z ) ≺ h ( z ) , g ( z ) z ≺ h ( z ) . For h ( z ) = (1 + z ) / (1 − z ) , and other appropriate choices of h , these functions haveearlier been studied, notably by MacGregor in [ – ], and Ratti in [ , ]. Recentinvestigations include those in [ , , ].In this paper, two specific choices of the function h are made: h ( z ) = √ z , and h ( z ) = e z . The Class T . This is the class given by T := (cid:26) f ∈ A : f ( z ) g ( z ) ≺ √ z, g ( z ) zp ( z ) ≺ √ z for some g ∈ A , p ( z ) ≺ √ z (cid:27) . This class is non-empty: let f , g , p : D → C be given by f ( z ) = z (1 + z ) / , g ( z ) = z (1 + z ) and p ( z ) = √ z. Then f ( z ) /g ( z ) ≺ √ z and g ( z ) /zp ( z ) ≺ √ z , so that f ∈ T . The function f will be shown to play the role of an extremal function for the class T . Since f ′ vanishes at z = − /
5, the function f is non-univalent, and thus, the class T contains non-univalentfunctions. Incidentally, f demonstrates the radius of univalence for T is at most 2 /
5. InTheorem 2.1, the radius of starlikeness for T is shown to be 2 /
5, whence T has radiusof univalence 2 / T . ADIUS OF STARLIKENESS 3
Lemma . Let p ( z ) ≺ √ z . Then p satisfies the sharp inequalities (1.1) √ − r ≤ | p ( z ) | ≤ √ r, | z | ≤ r, and (1.2) (cid:12)(cid:12)(cid:12)(cid:12) zp ′ ( z ) p ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ r − r ) , | z | ≤ r. Proof. If p ( z ) ≺ √ z , then p ( z ) = 1 + w ( z ) for some Schwarz function w . Thewell-known Schwarz lemma shows that | w ( z ) | ≤ | z | and(1.3) | w ′ ( z ) | ≤ − | w ( z ) | − | z | . Therefore, | p ( z ) | = | w ( z ) | ≤ | w ( z ) | ≤ | z | ≤ r for | z | ≤ r , that is, | p ( z ) | ≤ √ r for | z | ≤ r . Similarly, | p ( z ) | ≥ √ − r for | z | ≤ r .Since 2 zp ′ ( z ) /p ( z ) = zw ′ ( z ) / (1 + w ( z )), the inequality (1.3) readily shows2 (cid:12)(cid:12)(cid:12)(cid:12) zp ′ ( z ) p ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | z || w ′ ( z ) | − | w ( z ) | ≤ | z | (1 + | w ( z ) | )1 − | z | ≤ | z | (1 + | z | )1 − | z | = | z | − | z | ≤ r − r for | z | ≤ r . This proves (1.2). The inequalities are sharp for the function p ( z ) = √ z .For f ∈ T , let p ( z ) = f ( z ) /g ( z ) and p ( z ) = g ( z ) /zp ( z ). Then f ( z ) = zp ( z ) p ( z ) p ( z )and(1.4) (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) zp ′ ( z ) p ( z ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) zp ′ ( z ) p ( z ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) zp ′ ( z ) p ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . Since p, p , p ≺ √ z , we deduce from (1.2) and (1.4) that(1.5) (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ r − r ) , | z | ≤ r, for each function f ∈ T . Sharp growth inequalities also follow from (1.1): r (1 − r ) / ≤ | f ( z ) | ≤ r (1 + r ) / for each f ∈ T . Crude distortion inequalities can readily be obtained from (1.5) and thegrowth inequality; however, finding sharp estimates remain an open problem. The Class T . This class is defined by T := (cid:26) f ∈ A : f ( z ) g ( z ) ≺ e z , g ( z ) zp ( z ) ≺ e z for some g ∈ A , p ( z ) ≺ e z (cid:27) . Let f , g , p : D → C be given by f ( z ) = ze z , g ( z ) = ze z and p ( z ) = e z . Evidently, f ( z ) /g ( z ) ≺ e z , g ( z ) /zp ( z ) ≺ e z so that f ∈ T , and the class T is non-empty. Similar to f ∈ T , the function f plays the role of an extremal function for theclass T . The Taylor series expansion for f is f ( z ) = z + 3 z + 9 z z z · · · . R. M. ALI, K. SHARMA, AND V. RAVICHANDRAN
Comparing the second coefficient, it is clear that f is non-univalent. Hence the class T contains non-univalent functions. The derivative f ′ vanishes at z = − /
3, which showsthe radius of univalence for T is at most 1 /
3. From Theorem 2.1, the radius of starlikenessis shown to be 1 /
3, and so the radius of univalence for T is 1 / Lemma . Every p ( z ) ≺ e z satisfies the sharp inequalities (1.6) e − r ≤ | p ( z ) | ≤ e r , | z | ≤ r, and (1.7) (cid:12)(cid:12)(cid:12)(cid:12) zp ′ ( z ) p ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ r, | z | ≤ r ≤ √ − r ) − r ) , | z | = r ≥ √ − . Proof.
Let p ( z ) ≺ e z . Since p ( z ) = e w ( z ) for some Schwarz self-map w satisfying | w ( z ) | ≤ | z | , it follows that | p ( z ) | = e Re w ( z ) ≤ e | w ( z ) | ≤ e | z | . The function w also satisfy the sharp inequality (see [ , Corollary, p. 199])(1.8) | w ′ ( z ) | ≤ , r = | z | ≤ √ − r ) r (1 − r ) , r ≥ √ − . From zp ′ ( z ) /p ( z ) = zw ′ ( z ), we conclude that (cid:12)(cid:12)(cid:12)(cid:12) zp ′ ( z ) p ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ r, r = | z | ≤ √ − r ) − r ) , r ≥ √ − . This inequality is sharp for p ( z ) = e z and r = | z | ≤ √ −
1. It is also sharp in theremaining interval for the function p ( z ) = e w ( z ) , where w is the extremal function forwhich equality holds in (1.8).For f ∈ T , let p ( z ) = f ( z ) /g ( z ) and p ( z ) = g ( z ) /zp ( z ). Then f ( z ) = zp ( z ) p ( z ) p ( z )and(1.9) (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) zp ′ ( z ) p ( z ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) zp ′ ( z ) p ( z ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) zp ′ ( z ) p ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . Since p, p , p ≺ e z , estimates (1.7) and (1.9) show that(1.10) (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ r, r = | z | ≤ √ − r ) − r ) , r ≥ √ − . for each function f ∈ T . It also follows from (1.6) that re − r ≤ | f ( z ) | ≤ re r holds for each function f ∈ T , and that these estimates are sharp. ADIUS OF STARLIKENESS 5
In this paper, we shall adopt the commonly used notations for subclasses of A . First,for 0 ≤ α <
1, let S ∗ ( α ) denote the class of starlike functions of order α consisting offunctions f ∈ A satisfying the subordination zf ′ ( z ) f ( z ) ≺ − α ) z − z . Thus Re zf ′ ( z ) f ( z ) > α, z ∈ D . The case α = 0 corresponds to the classical functions whose image domains are starlikewith respect to the origin. Various other starlike subclasses of A occurring in the literaturecan be expressed in terms of the subordination(1.11) zf ′ ( z ) f ( z ) ≺ ϕ ( z )for suitable choices of the superordinate function ϕ . When ϕ : D → C is chosen tobe ϕ ( z ) := (1 + Az ) / (1 + Bz ), − ≤ B < A ≤
1, the subclass derived is denotedby S ∗ [ A, B ]. Functions f ∈ S ∗ [ A, B ] are known as Janowski starlike. When ϕ ( z ) :=1 + (2 /π )((log((1 + √ z ) / (1 − √ z ))) ), the subclass is denoted by S ∗ p , and its functionsare called parabolic starlike.In Section 2 of this paper, the radius of starlikeness, Janowski starlikeness, and par-abolic starlikeness are found for the classes T i , with i = 1 ,
2. Section 3 deals with thedetermination of the G -radius for the class T i with i = 1 ,
2, for certain other subclasses G occurring in the literature. These classes are associated with particular choices of the su-perordinate function ϕ in (1.11). As mentioned earlier, the G -radius for a given class M ,denoted by R G ( M ), is the largest number R such that r − f ( rz ) ∈ G for every 0 < r ≤ R and f ∈ M . It will become apparent in the forthcoming proofs that there are commonfeatures in the methodology of finding the G -radius for each of these subclasses.
2. Starlikeness of order α , Janowski and parabolic starlikeness The first result deals with the S ∗ ( α )-radius (radius of starlikeness of order α ) for theclasses T and T . This radius is shown to equal the S ∗ α -radius, where S ∗ α is the subclasscontaining functions f ∈ A satisfying | zf ′ ( z ) /f ( z ) − | < − α . The latter condition alsoimplies that S ∗ α ⊂ S ∗ ( α ). Theorem . Let ≤ α < . The radius of starlikeness of order α for T and T are(i) R S ∗ ( α ) ( T ) = R S ∗ α ( T ) = 2(1 − α ) / (5 − α ) ,(ii) R S ∗ ( α ) ( T ) = R S ∗ α ( T ) = (1 − α ) / . Proof. ( i ) The function σ ( r ) = (2 − r ) / (2 − r ) is a decreasing function on [0 , R := 2(1 − α ) / (5 − α ) is the root of the equation σ ( r ) = α . For f ∈ T and 0 < r ≤ R , the inequality (1.5) readily yieldsRe zf ′ ( z ) f ( z ) ≥ − r − r ) = 2 − r − r = σ ( r ) ≥ σ ( R ) = α and (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ r − r ) = 1 − σ ( r ) ≤ − σ ( R ) = 1 − α. R. M. ALI, K. SHARMA, AND V. RAVICHANDRAN At z = − R , the function f ∈ T given by f ( z ) = z (1 + z ) / yields zf ′ ( z ) f ( z ) = 2 + 5 z z = 2 − R − R = α. Thus Re zf ′ ( z ) f ( z ) = α and (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) = 1 − α. This proves that the S ∗ ( α ) and S ∗ α radii for T are the same number R .( ii ) Consider ω ( r ) = 1 − r , 0 ≤ r <
1. The number R = (1 − α ) / < / ω ( r ) = α . Since ω is decreasing, then ω ( r ) ≥ ω ( R ) = α for each f ∈ T and 0 < r ≤ R . It follows from (1.10) thatRe zf ′ ( z ) f ( z ) ≥ − r = ω ( r ) ≥ α, and (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ r = 1 − ω ( r ) ≤ − α. Evaluating the function f ( z ) = ze z at z = − R yields zf ′ ( z ) f ( z ) = 1 − R = α. Hence Re zf ′ ( z ) f ( z ) = α and (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) = 1 − α. This proves that the S ∗ ( α ) and S ∗ α radii for the class T are the same number R .Next we find the S ∗ [ A, B ]-radius (Janowski starlikeness) for T and T . Recall that S ∗ [ A, B ] consists of analytic functions f ∈ A satisfying the subordination zf ′ ( z ) /f ( z ) ≺ (1 + Az ) / (1 + Bz ), − ≤ B < A ≤ Theorem . (i) Every f ∈ T is Janowski starlike in the disc D r = { z : | z | < r } for r ≤ A − B ) / (3(1 + | B | ) + 2( A − B )) . If B < , then R S ∗ [ A,B ] ( T ) = 2( A − B ) / (3 + 2 A − B )) .(ii) The radius of Janowski starlikeness for T is R S ∗ [ A,B ] ( T ) = ( A − B ) / (3(1 − B )) . Proof.
Since S ∗ [ A, −
1] = S ∗ ((1 − A ) / B = − − < B < A ≤ i ) Let f ∈ T and write w = zf ′ ( z ) /f ( z ). Then (1.5) shows that | w − | ≤ r/ (2(1 − r ))for | z | ≤ r . For 0 ≤ r ≤ R := 2( A − B ) / (3(1 + | B | ) + 2( A − B )), then 3 R / ((2(1 − R )) =( A − B ) / (1 + | B | ).For 0 ≤ r ≤ R , we first show that the disc (cid:26) w : | w − | ≤ R − R ) = A − B | B | (cid:27) is contained in the images of the unit disc under the mapping (1 + Az ) / (1 + Bz ). As B = −
1, the image is the disc given by (cid:26) w : (cid:12)(cid:12)(cid:12)(cid:12) w − − AB − B (cid:12)(cid:12)(cid:12)(cid:12) < A − B − B (cid:27) . ADIUS OF STARLIKENESS 7
Silverman [ , p. 50-51] has shown that the disc { w : | w − c | < d } ⊂ { w : | w − a | < b } if and only if | a − c | ≤ b − d . With the choices c = 1, d = ( A − B ) / (1 + | B | ), a =(1 − AB ) / (1 − B ) and b = ( A − B ) / (1 − B ), then | a − c | = | B | ( A − B ) / (1 − B ) = b − d .This proves that S ∗ [ A, B ] radius is at least R .To prove sharpness, consider the function f ∈ T given by f ( z ) = z (1 + z ) / .Evidently, zf ′ ( z ) /f ( z ) = (2 + 5 z ) / (2 + 2 z ). For B <
0, evaluating at z = − R , then zf ′ ( z ) /f ( z ) = 1 + 3 z/ (2 + 2 z ) = 1 − ( A − B ) / (1 + | B | ) = (1 − A ) / (1 − B ). This showsthat (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − − AB − B (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − A − B − − AB − B (cid:12)(cid:12)(cid:12)(cid:12) = A − B − B , proving sharpness in the case B < ii ) Let f ∈ T and w := zf ′ ( z ) /f ( z ). It follows from (1.10) that | w − | ≤ r for | z | ≤ r . For 0 ≤ r ≤ R := ( A − B ) / (3(1 + | B | )), we see that the disc { w : | w − | ≤ R =( A − B ) / (1+ | B | ) } is contained in the disc { w : | w − (1 − AB ) / (1 − B ) | < ( A − B ) / (1 − B ) } ,as in the proof of (i). This proves that S ∗ [ A, B ] radius is at least R . The result is sharpfor the function f ∈ T given by the function f ( z ) = ze z .The function ϕ P AR : D → C given by ϕ P AR ( z ) := 1 + 2 π (cid:18) log 1 + √ z − √ z (cid:19) , Im √ z ≥ , maps D into the parabolic region ϕ P AR ( D ) = (cid:8) w = u + iv : v < u − (cid:9) = { w : Re w > | w − |} . The class C ( ϕ P AR ) = { f ∈ A : 1 + zf ′′ ( z ) /f ′ ( z ) ≺ ϕ P AR ( z ) } is the class of uniformlyconvex functions introduced by Goodman [ ]. The corresponding class S ∗ p := S ∗ ( ϕ P AR ) = { f ∈ A : zf ′ ( z ) /f ( z ) ≺ ϕ P AR ( z ) } introduced by Rønning [ ] is known as the class ofparabolic starlike functions. The class S ∗ p consists of functions f ∈ A satisfyingRe (cid:18) zf ′ ( z ) f ( z ) (cid:19) > (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) , z ∈ D . Evidently, every parabolic starlike function is also starlike of order 1/2. The radius ofparabolic starlikeness for the class T and T is given in the next result. Corollary . The radius of parabolic starlikeness for T and T is respectivelyequal to its radius of starlikeness of order / . Thus,(i) R S ∗ p ( T ) = 1 / ,(ii) R S ∗ p ( T ) = 1 / . Proof.
Shanmugam and Ravichandran [ , p. 321] proved that { w : | w − a | < a − / } ⊆ { w : Re w > | w − |} for 1 / < a ≤ /
2. Choosing a = 1, this implies that S ∗ / ⊂ S ∗ p . Every parabolicstarlike function is also starlike of order 1/2, whence the inclusion S ∗ / ⊂ S ∗ p ⊂ S ∗ (1 / F , readily R S ∗ / ( F ) ≤ R S ∗ p ( F ) ≤ R S ∗ (1 / ( F ). R. M. ALI, K. SHARMA, AND V. RAVICHANDRAN
When F = T i , i = 1 ,
2, Theorem 2.1 gives R S ∗ ( α ) ( T i ) = R S ∗ α ( T i ). This shows that R S ∗ / ( T i ) = R S ∗ p ( T i ) = R S ∗ (1 / ( T i ). Since R S ∗ (1 / ( T ) = 1 / R S ∗ (1 / ( T ) = 1 / R S ∗ p ( T ) = 1 / R S ∗ p ( T ) = 1 /
3. Further radius of starlikeness
In this section, we find the G -radius for the class T i with i = 1 ,
2, for certain otherwidely studied subclasses G . These are associated with particular choices of the superor-dinate function ϕ in (1.11).Denote by S ∗ exp := S ∗ ( e z ) the class associated with ϕ ( z ) := e z in (1.11). This classwas introduced by Mendiratta et al. [ ], and it consists of functions f ∈ A satisfyingthe condition | log( zf ′ ( z ) /f ( z )) | <
1. The following result gives the radius of exponentialstarlikeness for the classes T and T . Corollary . The S ∗ exp -radius for the class T is R S ∗ exp ( T ) = (2 − e ) / (2 − e ) ≈ . , while that of T is R S ∗ exp ( T ) = ( e − / e. Proof.
Mendiratta et al. [ , Lemma 2.2] proved that { w : | w − a | < a − /e } ⊆ { w : | log w | < } for e − ≤ a ≤ ( e + e − ) /
2, and this inclusion with a = 1 gives S ∗ /e ⊂ S ∗ exp . It was alsoshown in [ , Theorem 2.1 (i)] that S ∗ exp ⊂ S ∗ (1 /e ). Therefore, S ∗ /e ⊂ S ∗ exp ⊂ S ∗ (1 /e ) , which, as a consequence of Theorem 2.1, established the result.Corollary 3.2 investigates the radius of cardioid starlikeness for each class T and T .The class S ∗ C := S ∗ ( ϕ CAR ), where ϕ CAR ( z ) = 1 + 4 z/ z / , – ]. Descriptively, f ∈ S ∗ C provided zf ′ ( z ) /f ( z ) lies in the regionbounded by the cardioid Ω C := { w = u + iv : (9 u +9 v − u +5) − u +9 v − u +1) =0 } . Corollary . The following are the S ∗ C -radius for the classes T and T :(i) R S ∗ C ( T ) = 4 / ,(ii) R S ∗ C ( T ) = 2 / . Proof.
Sharma et al. [ ] proved that { w : | w − a | < a − / } ⊆ Ω C for 1 / < a ≤ /
3, and this inclusion with a = 1 gives S ∗ / ⊂ S ∗ C . Thus R S ∗ / ( T i ) ≤ R S ∗ C ( T i ) for i = 1 , R S ∗ C ( T i ) ≤ R S ∗ / ( T i ) for i = 1 , i ) Evaluating the function f ( z ) = z (1 + z ) / at z = − R = − R S ∗ / ( T ) = − / zf ′ ( z ) f ( z ) = 2 + 5 z z = 2 − R − R = 13 = ϕ CAR ( − . Thus, R S ∗ C ( T ) ≤ / ii ) Similarly, at z = − R = − R S ∗ / ( T ) = − /
9, the function f ( z ) = ze z yields zf ′ ( z ) f ( z ) = 1 + 3 z = 1 − R = 13 = ϕ CAR ( − . ADIUS OF STARLIKENESS 9
This proves that R S ∗ C ( T ) ≤ / . In 2019, Cho et al. [ ] studied the class S ∗ sin := S ∗ (1 + sin z ) consisting of functions f ∈ A satisfying the condition zf ′ ( z ) /f ( z ) ≺ z . We find the S ∗ sin -radius for theclasses T and T . Corollary . The following are the S ∗ sin -radius for each class T and T :(i) R S ∗ sin ( T ) = 2(sin 1) / (3 + 2 sin 1) ≈ . .(ii) R S ∗ sin ( T ) = (sin 1) / . Proof.
It was proved in [ ] that { w : | w − a | < sin 1 −| a − |} ⊆ q ( D ) for | a − | ≤ sin 1,where q ( z ) := 1 + sin z . For a = 1, this implies that S ∗ − sin 1 ⊂ S ∗ sin . Thus R S ∗ − sin 1 ( T i ) ≤ R S ∗ sin ( T i ) for i = 1 ,
2. The proof is completed by demonstrating R S ∗ sin ( T i ) ≤ R S ∗ − sin 1 ( T i )for i = 1 , i ) Evaluating the function f ( z ) = z (1 + z ) / at z = − R = − R S ∗ − sin 1 ( T ) = − / (3 + 2 sin 1) gives zf ′ ( z ) f ( z ) = 2 + 5 z z = 2 − R − R = 1 − sin 1 = q ( − . Thus, R S ∗ sin ( T ) ≤ / (3 + 2 sin 1).( ii ) Similarly, at z = ± R = ± R S ∗ − sin 1 ( T ) = ± (sin 1) /
3, the function f ( z ) = ze z yields zf ′ ( z ) f ( z ) = 1 + 3 z = 1 ± R = 1 ± sin 1 = q ( ± . This proves that R S ∗ sin ( T ) ≤ (sin 1) / . Consider next the class S ∗ $ := S ∗ ( z + √ z ) introduced by Raina and Sok´o l in[ ]. Functions f ∈ S ∗ $ provided zf ′ ( z ) /f ( z ) lies in the region bounded by the luneΩ l := { w : | w − | < | w |} . The result below gives the radius of lune starlikeness foreach class T and T . Corollary . The following are the S ∗ $ -radius for each class T and T :(i) R S ∗ $ ( T ) = 2( √ − / (2 √ − ≈ . .(ii) R S ∗ $ ( T ) = (2 − √ / . Proof.
It was shown by Gandhi and Ravichandran [ , Lemma 2.1] that { w : | w − a | < − |√ − a |} ⊆ Ω l for √ − < a ≤ √ a = 1, the inclusion gives S ∗√ − ⊂ S ∗ $ . Thus R S ∗√ − ( T i ) ≤ R S ∗ $ ( T i ) for i = 1 ,
2. We complete the proof bydemonstrating R S ∗ $ ( T i ) ≤ R S ∗√ − ( T i ) for i = 1 , i ) Evaluating the function f ( z ) = z (1 + z ) / at z = − R = − R S ∗√ − ( T ) = − √ − / (2 √ −
7) gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) zf ′ ( z ) f ( z ) (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) z z (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − R − R (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 .
828 = 2 (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . Thus, R S ∗ $ ( T ) ≤ √ − / (2 √ − ( ii ) Similarly, at z = − R = − R S ∗√ − ( T ) = − (2 − √ /
3, the function f ( z ) = ze z yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) zf ′ ( z ) f ( z ) (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | (1 + 3 z ) − | = | (1 − R ) − | = 0 .
828 = 2 (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . This proves that R S ∗ $ ( T ) ≤ (2 − √ / . As a further example, consider next the class S ∗ R := S ∗ ( η ( z )), where η ( z ) = 1 + (( zk + z ) / ( k − kz )), k = √ ]. Corollary . The following are the S ∗ R -radius for the classes T and T :(i) R S ∗ R ( T ) = 2( − √ / (4 √ − ≈ . ,(ii) R S ∗ R ( T ) = (3 − √ / . Proof.