Radius of starlikeness for some classes containing non-univalent functions
aa r X i v : . [ m a t h . C V ] J a n RADIUS OF STARLIKENESS FOR SOME CLASSES CONTAININGNON-UNIVALENT FUNCTIONS
SHALU YADAV, KANIKA SHARMA, AND V. RAVICHANDRAN
Abstract.
A starlike univalent function f is characterized by the function zf ′ ( z ) /f ( z );several subclasses of these functions were studied in the past by restricting the function zf ′ ( z ) /f ( z ) to take values in a region Ω on the right-half plane, or, equivalently, by requiringthe function zf ′ ( z ) /f ( z ) to be subordinate to the corresponding mapping of the unit disk D to the region Ω. The mappings w ( z ) := z + √ z , w ( z ) := √ z and w ( z ) := e z maps the unit disk D to various regions in the right half plane. For normalized analyticfunctions f satisfying the conditions that f ( z ) /g ( z ) , g ( z ) /zp ( z ) and p ( z ) are subordinateto the functions w i , i = 1 , , g ( z ) and p ( z ), wedetermine the sharp radius for them to belong to various subclasses of starlike functions. Introduction
Though complex numbers is not ordered field, the inequalities in the real line can beextended to complex plane in a natural way using the concept of subordination. Let f, g betwo analytic functions defined on the open unit disk D := { z ∈ C : | z | < } . The function f is subordinate to the function g , written f ≺ g , if f = g ◦ w for some analytic function w : D → D with w (0) = 0 (and such a function w is known as a Schwartz function). Aunivalent function f : D → C is always locally univalent or, in other words, it has non-vanishing derivative. Therefore, the study of univalent functions can be restricted to thefunctions normalized by f (0) = f ′ (0) − A denote the class of all analyticfunctions f : D → C normalized by the conditions f (0) = 0 and f ′ (0) = 1. Since f ′ (0) = 0for functions f ∈ A , the functions in the class A are univalent in some disk centred atthe origin. The largest disk centered at the origin in which f is univalent is called as theradius of univalence of the function f . Consider a subset M ⊂ A and a property P (suchas univalence, starlikeness, convexity) that the image of the functions in M may or maynot have. It often happens that the image of D r = { z ∈ D : | z | < r } for some r ≤ P ; the largest ρ f such that the functions has the property P in eachdisk D r for 0 < r ≤ ρ f is the radius of the property P of the function f . The number ρ := inf { ρ f : f ∈ M} is the radius of the property P of the class M ; if G is the class of all f ∈ A characterized by the property P , then ρ is called the G -radius of the class M . The G -radius of the class M is denoted by R G ( M ) or simply by R G if the class M is clear fromthe context.The class M we are interested in is characterized by the ratio between functions f and g belonging to A ; several authors [1, 2, 6, 9, 10, 11, 15, 17] have studied such classes. The Mathematics Subject Classification.
Key words and phrases.
Starlike functions; subordination; radius of starlikeness; lune.The research of Shalu Yadav is supported by an institute fellowship from NIT Trichy. classes which we are considering are as follows. T = { f ∈ A : fg ≺ e z , gzp ≺ e z for some g ∈ A and p ≺ √ z }T = { f ∈ A : fg ≺ √ z, gzp ≺ √ z for some g ∈ A and p ≺ e z }T = { f ∈ A : fg ≺ z + √ z , gzp ≺ z + √ z for some g ∈ A and p ≺ z + √ z } . These classes are motivated by a recent work of Ali, Sharma and Ravichandran [ ? ] whereinsimilar classes were investigated. These classes contain non-univalent functions and thismakes the study of such functions interesting. We compute G -radius when G is one ofthe following subclasses of starlike functions studied recently in the literature. A starlikeunivalent function f is characterized by the condition Re( zf ′ ( z ) /f ( z )) >
0. If we define theclass P as the class of all functions p ( z ) = 1 + c z + · · · defined on D satisfying Re p ( z ) > f is starlike if and only if zf ′ ( z ) /f ( z ) ∈ P . Several subclasses ofstarlike functions were studied in the past by restricting the function zf ′ ( z ) /f ( z ) to takevalues in a region Ω on the right-half plane, or, equivalently, by requiring the function zf ′ ( z ) /f ( z ) to be subordinate to the corresponding mapping ϕ : D → Ω: zf ′ ( z ) /f ( z ) ≺ ϕ ( z ). For ϕ ( z ) = (1 + (1 − α ) z ) / (1 − z ), 0 ≤ α <
1, this class is the class S ∗ ( α ) ofall starlike functions of order α . Several other subclasses of starlike functions are definedby replacing the superordinate functions ϕ by functions having nice geometry. For thefunctions ϕ defined by ϕ ( z ) := (1 + Az ) / (1 + Bz ), with − ≤ B < A ≤ e z , / z +(2 / z , z, z + √ z , zk + z ) / ( k − kz ) where k = √ √ z ) / (1 − √ z )) π ) , / (1 + e − z ) and 1 + z − z /
3, we denote the class of all functions f ∈ A with zf ′ ( z ) /f ( z ) ≺ ϕ ( z ) respectively by S ∗ [ A, B ] , S ∗ exp , S ∗ c , S ∗ sin , S ∗ $ S ∗ R , S ∗ p , S ∗ SG and S ∗ N e .2. The radius of univalence
Class T . This class consists of the analytic functions f such that f ( z ) /g ( z ) is sub-ordinate to e z and p is subordinate to √ z for some analytic function g . This class isnon-empty as the function f : D → C , defined by f ( z ) = ze z √ z belongs to the class T . The function f satisfies the subordination condition for this classalong with the functions g , p : D → C by g ( z ) = ze z √ z and p ( z ) = √ z. The function f plays the role of extremal for this class. Since f ′ ( z ) = e z (4 z + 7 z + 2)2 √ z , it is clear that f ′ ( − √ /
8) = 0, and so the radius of univalence is at most ( − √ / f is not univalent in D , the class T contains non-univalent functions. Herealso we find that radius of univalence and the radius of starlikeness for T are same and is( − √ / ≈ − . ADIUS OF STARLIKENESS 3
To do this, we need to first find a disk into which the disk D r is mapped by the function f ∈ T . Let f ∈ T and define the functions p , p by p ( z ) = f ( z ) /g ( z ) and p ( z ) = g ( z ) /zp ( z ). Then f ( z ) = zp ( z ) p ( z ) p ( z ) and (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) . (2.1)Using the bounds for p, p , p , we obtain (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( r (5 − r )2(1 − r ) ; r ≤ √ − r +7 r +3 r − r ) ; r ≥ √ − . (2.2)for each function f ∈ T . Clearly, for | z | = r ≤ ( − √ /
8, we have (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ r (5 − r )2(1 − r ) ≤ . This shows that the radius of starlikeness is at least ( − √ /
8. Since the radius ofunivalence is atmost ( − √ /
8, it follows that the radius univalence and radius ofstarlikeness are both equal to ( − √ / Class T . Functions of this class are analytic functions f satisfying the subordinations f ( z ) g ( z ) ≺ √ z, g ( z ) zp ( z ) ≺ √ z and p ( z ) ≺ e z , where the functions g ∈ A and p ∈ P .This class is non-empty as the function f : D → C defined by f ( z ) = z (1 + z ) e z belongs to the class T . The function f satisfies the required subordinations when we definethe functions g , p : D → C by g ( z ) = ze z √ z and p ( z ) = e z . This function f plays the role of extremal for this class. Since f ′ ( z ) = e z (1 + 3 z + z ) , it is clear that f ′ ( − √ /
2) = 0, and so the radius of univalence is atmost ( − √ / T are equaland commom value of the radius is precisely ( − √ / ≈ − . D r is mapped by the function f ∈ T . Let f ∈ T and define the functions p , p by p ( z ) = f ( z ) /g ( z ) and p ( z ) = g ( z ) /zp ( z ). Then f ( z ) = zp ( z ) p ( z ) p ( z ) and (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) . (2.3)Using the bounds for p, p , p , we obtain (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( r ( − r )( − r ) ; r ≤ √ − r +6 r + r − r ) ; r ≥ √ − . (2.4)for each function f ∈ T . Clearly, for | z | = r ≤ ( − √ /
2, we have
S. YADAV, K. SHARMA, AND V. RAVICHANDRAN (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ r ( − r )( − r ) ≤ . This shows that the radius of starlikeness is at least ( − √ /
2. Since the radius of univa-lence is atmost ( − √ /
2, it follows that the radius univalence and radius of starlikenessare both equal to ( − √ / Class T . Recall that a function f ∈ A belongs the class T if there are functions g ∈ A and p ∈ P satisfying the subordinations f ( z ) g ( z ) ≺ z + √ z , g ( z ) zxp ( z ) ≺ z + √ z and p ( z ) ≺ z + √ z . This class is non-empty as the function f : D → C defined by f ( z ) = z ( z + √ z ) belongs to the class T . Indeed, the function f satisfies the required subordinations whenwe define the functions g , p : D → C by g ( z ) = z ( z + √ z ) and p ( z ) = z + √ z . The function f plays the role of extremal for this class. Since f ′ ( z ) = (cid:0) z + √ z (cid:1) (cid:0) z + √ z (cid:1) √ z , it is clear that f ′ ( − / √
8) = 0, and so the radius of univalence is at most 1 / √
8. Also, sincethe function f is not univalent in D , the class T contains non-univalent functions. Indeed,we shall show that the radius of univalence and the radius of starlikeness for T are equaland the common value of the radius is precisely 1 / √ ≈ . D r is mapped by the function f ∈ T . Let f ∈ T and define the functions p , p by p ( z ) = f ( z ) /g ( z ) and p ( z ) = g ( z ) /zp ( z ). Then f ( z ) = zp ( z ) p ( z ) p ( z ) and (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) . (2.5)For p ∈ P with p ( z ) ≺ z + √ z , Afis and Noor [4] have shown that | p ( z ) | ≤ r + √ r , (cid:12)(cid:12)(cid:12)(cid:12) zp ′ ( z ) p ( z ) ≤ r √ r (cid:12)(cid:12)(cid:12)(cid:12) ( | z | ≤ r ) . Using these inequalities for p, p , p ≺ z + √ z , we see that (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ r √ r , | z | ≤ r, (2.6)for each function f ∈ T . Clearly, for | z | = r ≤ / √
8, we have (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ r √ r ≤ . ADIUS OF STARLIKENESS 5
This shows that the radius of starlikeness is at least 1 / √
8. Since the radius of univalenceis at most 1 / √
8, it follows that the radius of univalence and the radius of starlikeness areboth equal to 1 / √
8. 3.
Radius of starlikeness
Our first theorem gives the sharp radius of starlikeness of order α of the classes T , T and T . We shall show that this radius is the same for the subclass S ∗ α consisting of all functions f ∈ S ∗ ( α ) satisfying | zf ′ ( z ) /f ( z ) − | < − α . Theorem 3.1.
The following results hold for the classes S ∗ ( α ) and S ∗ α .(i) R S ∗ ( α ) ( T ) = R S ∗ α ( T ) = (7 − α − √
17 + 4 α + 4 α ) / (ii) R S ∗ ( α ) ( T ) = R S ∗ α ( T ) = ((3 − α − √ − α + α )) / (iii) R S ∗ ( α ) ( T ) = R S ∗ α ( T ) = (1 − α ) / ( √ α − α ) Proof. ( i ) The function defined by m ( r ) = (4 r − r + 2) / − r ) , ≤ r < ρ = R S ∗ α ( T ) is the root of the equation m ( r ) = α . From (2.2), it follows thatRe zf ′ ( z ) f ( z ) ≥ r − r + 22(1 − r ) = m ( r ) ≥ m ( ρ ) = α. This shows that R S ∗ α ( T ) is atleast ρ . At z = − R S ∗ α ( T ) = − ρ , the function f satisfies zf ′ ( z ) f ( z ) = 4 ρ − ρ + 22(1 − ρ ) = α. Thus the result is sharp.Also, (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 radii R S ∗ ( α ) ( T ) and R S ∗ α ( T ) are same.( ii ) The function defined by n ( r ) = ( r − r + 1) / (1 − r ) , ≤ r < ρ = R S ∗ α ( T ) is the root of the equation n ( r ) = α . From (2.4), it follows thatRe zf ′ ( z ) f ( z ) ≥ − r + r − r = n ( r ) ≥ n ( ρ ) = α. This shows that R S ∗ α ( T ) is atleast ρ . At z = − R S ∗ α ( T ) = − ρ , the function f satisfies zf ′ ( z ) f ( z ) = 1 − ρ + ρ − ρ = α. Also, for the function f we have (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) = 1 − α. Hence, the radii R S ∗ ( α ) and R S ∗ α are same for the class T .( iii ) The function s ( r ) = 1 − r/ √ r , ≤ r < ρ = R S ∗ α ( T ) is the root of the equation s ( r ) = α . From (2.6), it follows S. YADAV, K. SHARMA, AND V. RAVICHANDRAN Re (cid:18) zf ′ ( z ) f ( z ) (cid:19) ≥ − r √ r = s ( r ) ≥ s ( ρ ) = α. This shows that R S ∗ α ( T ) is atleast ρ . At z = − R S ∗ α ( T ) = − ρ . For the function f ( z ), wehave zf ′ ( z ) f ( z ) = 1 − ρ p ρ = α. Now, considering (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ r √ r ≤ − α. The function f ( z ) provides sharpness, as (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) = 3 z √ z = 1 − α. The radii R S ∗ ( α ) and R S ∗ α are same for the class T .The function ϕ P AR : D → C given by ϕ P AR := 1 + 2 π (cid:18) log 1 + √ z − √ z (cid:19) , Im √ z ≥ D on the parabolic region given by ϕ P AR ( D ) = { w = u + iv : v < u − { w :Re w > | w − |} . The class S ∗ p := S ∗ ( ϕ P AR ) = { f ∈ A : zf ′ ( z ) /f ( z ) ≺ ϕ P AR ( z ) } wasintroduced by Rønning [14], and is known as the class of parabolic starlike functions. Theclass S ∗ p consists of the 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 / / < a < /
2, then { w : | w − a | < a − / } ⊂ { w : Re w > | w − |} . (3.1)The following result gives the radius of parabolic starlikeness for the classes T , T and T . Corollary 3.2.
The following results holds for the class S ∗ p .(i) R S ∗ p ( T ) = (3 − √ / ≈ . (ii) R S ∗ p ( T ) = (5 − √ / ≈ . (iii) R S ∗ p ( T ) = 1 / √ ≈ . Proof.
In equation (3.1), putting a = 1 gives S ∗ / ⊂ S ∗ p . Every parabolic starlike functionis also starlike function 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 ).When F = T i , i = 1 , ,
3, Theorem 3 . 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 ). So for α = 1 /
2, from Theorem 3 .
1, it follows that R S ∗ p ( T ) = (3 − √ / , R S ∗ p ( T ) = (5 − √ / R S ∗ p ( T ) = 1 / √ ADIUS OF STARLIKENESS 7 ( i ) For the function f ( z ) = ze z √ z , at z = − R S ∗ p ( T ) = − ρ , we haveRe zf ′ ( z ) f ( z ) = 4 ρ − ρ + 22(1 − ρ ) = 12 = (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) . Thus, R S ∗ p ( T ) ≤ (3 − √ / ii ) For the function f ( z ) = z (1 + z ) e z , at z = − R S ∗ p ( T ) = − ρ , we obtainRe zf ′ ( z ) f ( z ) = ρ + 3 ρ + 1(1 + ρ ) = 12 = (cid:12)(cid:12)(cid:12)(cid:12) zf ′ ( z ) f ( z ) − (cid:12)(cid:12)(cid:12)(cid:12) . Thus R S ∗ p ( T ) ≤ (5 − √ / iii ) For the function f ( z ) = z ( z + √ z ) , at z = − R S ∗ p ( T ) = − ρ , we haveRe zf ′ ( z ) f ( z ) = 1 − ρ p ρ = 12 = (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 ∗ p ( T ) ≤ / √ et al. [12] introduced the class of starlike functions associated withthe exponential function as S ∗ e = S ∗ ( e z ) and it satisfies the condition | log zf ′ ( z ) /f ( z ) < | .They had also proved that, for e − ≤ a ≤ ( e + e − / { w ∈ C : | w − a | < a − e − } ⊆ { w ∈ C : | log w | < } . (3.2) Corollary 3.3.
The following results hold for the class S ∗ e .(i) R S ∗ e ( T ) = ( − e − √ e + 17 e ) / e ≈ . (ii) R S ∗ e ( T ) = ( − e − √ − e + 5 e ) / e ≈ . (iii) R S ∗ e ( T ) = (1 − e + e ) / ( − e + 8 e ) ≈ . Proof.
Mendiratta et. al provided the inclusion (3.2), which gives S ∗ /e ⊂ S ∗ e . It was alsoshown in [12, Theorem 2.1(i)] that S ∗ e ⊂ S ∗ /e . Therefore, S ∗ /e ⊂ S ∗ e ⊂ S ∗ (1 /e ), whichprovides the required radii as a consequence of Theorem 3 . i ) For the function f ( z ) = ze z √ z , at z = − R S ∗ e ( T ) = − ρ , we have (cid:12)(cid:12)(cid:12)(cid:12) log zf ′ ( z ) f ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) log 4 ρ − ρ + 22(1 − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 . Thus, R S ∗ e ( T ) ≤ ( − e − √ e + 17 e ) / e .( ii ) For the function f ( z ) = z (1 + z ) e z , at z = − R S ∗ e ( T ) = − ρ , we get (cid:12)(cid:12)(cid:12)(cid:12) log zf ′ ( z ) f ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) log 1 − ρ + ρ (1 − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 . Thus, R S ∗ e ( T ) ≤ ( − e − √ − e + 5 e ) / e .( iii ) For the function f ( z ) = z ( z + √ z ) , at z = − R S ∗ e ( T ) = − ρ , we obtain (cid:12)(cid:12)(cid:12)(cid:12) log zf ′ ( z ) f ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log − ρ p ρ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 . This proves that R S ∗ e ( T ) ≤ (1 − e + e ) / ( − e + 8 e ). S. YADAV, K. SHARMA, AND V. RAVICHANDRAN
Sharma et. al [17] studied the various properties of the class S ∗ c = S ∗ (1+(4 / z +(2 / z .This class consists the functions f such that zf ′ ( z ) /f ( z ) lies in the region bounded by thecardioid Ω c = { u + iv : (9 u + 9 v − u + 5) − u + 9 v − u + 1) = 0 } . They provedthe result that, for 1 / < a ≤ / { w ∈ C : | w − a | < (3 a − / } ⊆ Ω c . (3.3)Various results related to this class are investigated in these papers [13, 16, 18, 19]. Followingcorollary provides the radius of cardioid starlikeness for each class T , T and T . Corollary 3.4.
The following result holds for the class S ∗ c .(i) R S ∗ c ( T ) = 1 / . (ii) R S ∗ c ( T ) = (4 − √ / ≈ . (iii) R S ∗ c ( T ) = 2 / √ ≈ . Proof.
Equation (3.3) provides the inclusion S ∗ / ⊂ S ∗ c for a = 1. Thus R S ∗ / ( T i ) ≤ R S ∗ c ( T i )for i = 1 , ,
3. The proof is completed by demonstrating R S ∗ c ( T i ) ≤ R S ∗ / ( T i ) for i = 1 , , i ) For the function f ( z ) = ze z √ z , at z = − R S ∗ c ( T ) = − ρ , we obtain (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) ρ − ρ + 22(1 − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) = 13 . Thus, R S ∗ c ( T ) ≤ / ii ) For the function f ( z ) = z (1 + z ) e z , at z = − R S ∗ c ( T ) = − ρ , we have (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) − ρ + ρ (1 − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) = 13 . Thus, R S ∗ c ( T ) ≤ (4 − √ / iii ) For the function f ( z ) = z ( z + √ z ) , at z = − R S ∗ c ( T ) = − ρ , (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)(cid:12) − ρ p ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 13 . This proves that R S ∗ c ( T ) ≤ / √ et al. [5] considered the class of starlike functions associated with sinefunction. The class S ∗ sin is defined as S ∗ sin = { f ∈ A : zf ′ ( z ) /f ( z ) ≺ z := q ( z ) } for z ∈ D . For | a − | ≤ sin 1, the following inclusion holds { w ∈ C : | w − a | < sin 1 − | a − |} ⊆ Ω s . (3.4)Here Ω s := q ( D ) is the image of the unit disk D under the mapping q ( z ) = 1 + sin z .We find the S ∗ sin - radius for the classes T , T and T . Corollary 3.5.
The following results hold for the class S ∗ sin .(i) R S ∗ sin ( T ) = (5 + 2 sin 1 − √ −
12 sin 1 + 4 sin 1 ) / ≈ . (ii) R S ∗ sin ( T ) = (2 + sin 1 − √ ) / ≈ . (iii) R S ∗ sin ( T ) = sin 1 / √ − sin 1 ≈ . ADIUS OF STARLIKENESS 9
Proof.
By putting a = 1 in equation (3.4) we obtain the inclusion S ∗ − sin ⊂ S ∗ sin . Thus R S ∗ − sin ≤ R S ∗ sin for i = 1 , ,
3. The proof is completed by demonstrating R S ∗ − sin ≤ R S ∗ sin for i = 1 , ,
3. ( i ) For the function f ( z ) = ze z √ z , at z = − R S ∗ sin ( T ) = − ρ , we get (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) ρ − ρ + 22(1 − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 + sin 1 . Thus, R S ∗ sin ( T ) ≤ (5 + 2 sin 1 − √ −
12 sin 1 + 4 sin 1 ) / ii ) For the function f ( z ) = z (1 + z ) e z , at z = − R S ∗ sin ( T ) = − ρ , we have (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) − ρ + ρ (1 − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 + sin 1 . Thus, R S ∗ sin ( T ) ≤ (2 + sin 1 − √ ) / iii ) For the function f ( z ) = z ( z + √ z ) , at z = − R S ∗ sin ( T ) = − ρ , we get (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)(cid:12) − ρ p ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 + sin 1 . This proves that R S ∗ sin ( T ) ≤ sin 1 / √ − sin 1 .In the next result, we find the radius for starlike functions accociated with a rationalfunction. Kumar and Ravichandran [8] introduced the class of starlike functions associatedwith a rational function, ψ ( z ) = 1 + ( z k + z ) / ( k − kz )) where k = √ S ∗ R = S ∗ ( ψ ( z )). For 2( √ − < a ≤ √
2, they had proved that { w ∈ C : | w − a | < a − √ − } ⊆ ψ ( D ) . (3.5) Corollary 3.6.
The following results holds for the class S ∗ R .(i) R S ∗ R ( T ) = (11 − √ − p − √ / ≈ . (ii) R S ∗ R ( T ) = (5 − √ − p − √ / ≈ . (iii) R S ∗ R ( T ) = ( p −
38 + 27 √ / √ ≈ . Proof.
For a = 1 equation (3.5) gives the inclusion S ∗ √ − ⊂ S ∗ R . Thus R S ∗ √ − ( T i ) ≤ R S ∗ R ( T i ) for i = 1 , ,
3. We next show that R S ∗ R ( T i ) ≤ R S ∗ √ − ( T i ) for i = 1 , , i ) For the function f ( z ) = ze z √ z , at z = − R S ∗ R ( T ) = − ρ , we get (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) ρ − ρ + 22(1 − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) = 2( √ − . Thus, R S ∗ R ( T ) ≤ (11 − √ − p − √ / ii ) For the function f ( z ) = z (1 + z ) e z , at z = − R S ∗ R ( T ) = − ρ , we have (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) − ρ + ρ (1 − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) = 2( √ − . Thus, R S ∗ R ( T ) ≤ (5 − √ − p − √ / ( iii ) For the function f ( z ) = z ( z + √ z ) , at z = − R S ∗ R ( T ) = − ρ , we obtain (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)(cid:12) − ρ p ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 2( √ − . Thus, R S ∗ R ( T ) ≤ ( p −
38 + 27 √ / √ S ∗ Ne = S ∗ (1 + z − ( z / D onto the interior of a two cusped kidney shaped curve Ω Ne := { u + iv :(( u − + v − / − v / < } . For 1 / < a ≤
1, they had proved that { w ∈ C : | w − a | < a − / } ⊆ Ω Ne . (3.6)Our next theorem determines the S ∗ Ne radius results for the classes T , T and T . Corollary 3.7.
The following results hold for the class S ∗ Ne .(i) R S ∗ Ne ( T ) = 1 / . (ii) R S ∗ Ne ( T ) = (4 − √ / ≈ . (iii) R S ∗ Ne ( T ) = 2 / √ ≈ . Proof.
From equation (3.6) we obtain the inclusion S ∗ / ⊂ S ∗ Ne for a = 1. This shows that R S ∗ / ( T i ) ≤ R S ∗ Ne ( T i ) for i = 1 , , i ) For the function f ( z ) = ze z √ z , at z = − R S ∗ Ne ( T ) = − ρ , (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) ρ − ρ + 22(1 − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) = 13 . Thus, R S ∗ Ne ( T ) ≤ / ii ) For the function f ( z ) = z (1 + z ) e z , at z = − R S ∗ R ( T ) = − ρ , (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) − ρ + ρ (1 − ρ ) (cid:12)(cid:12)(cid:12)(cid:12) = 13 . Thus, R S ∗ Ne ( T ) ≤ (4 − √ / iii ) For the function f ( z ) = z ( z + √ z ) , at z = − R S ∗ Ne ( T ) = − ρ , (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)(cid:12) − ρ p ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 13 . This proves that R S ∗ Ne ( T ) ≤ / √ S ∗ SG := S ∗ (2 / e − z ), where 2 / (1 + e − z ) is amodified sigmoid function that maps D onto the region Ω SG := { w = u + iv : | log( w/ (2 − w )) | < } . Precisely, f ∈ S ∗ SG provided the function zf ′ ( z ) /f ( z ) maps D onto the regionlying inside the domain Ω SG . For 2 / ( e + 1) < a < e/ (1 + e ), Goel and Kumar [7] hadproved the following inclusion { w ∈ C : | w − a | < r SG } ⊂ Ω SG , (3.7)provided r SG = (( e − / ( e + 1)) − | a − | . Next result is about the S ∗ SG radius for thedefined classes. ADIUS OF STARLIKENESS 11
Theorem 3.8.
The following results hold for the class S ∗ SG .(i) R S ∗ SG ( T ) = (3 + 7 e − √
41 + 42 e + 17 e ) / (8 + 8 e ) ≈ . (ii) R S ∗ SG ( T ) = (1 + 3 e − √ e + 5 e )(2 + 2 e ) ≈ . (iii) R S ∗ SG ( T ) = ( √ − e + e ) / (8 + 20 e + 8 e ) ≈ . Proof. ( i ) The function defined by m ( r ) = (4 r − r + 2) / − r ) , ≤ r < ρ = R S ∗ SG ( T ) is the root of the equation m ( r ) = 2 / (1 + e ). For 0 < r ≤ R S ∗ SG ( T ), we have m ( r ) ≥ / (1 + e ). That is r (5 − r )2(1 − r ) ≤ e − e + 1 . For the class T , the centre of the disk is 1, therefore the disk obtained in (2.2) is containedin the region bounded by modified sigmoid, by equation (3.7). For the function f ( z ) = ze z √ z , at z = − R S ∗ SG ( T ) = − ρ , we have (cid:12)(cid:12)(cid:12)(cid:12) log zf ′ ( z ) /f ( z )2 − ( zf ′ ( z ) /f ( z )) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (4 ρ − ρ + 2) / − ρ )2 − ((4 ρ − ρ + 2) / − ρ )) (cid:12)(cid:12)(cid:12)(cid:12) = 1 . ( ii ) The function defined n ( r ) = ( r − r + 1) / (1 − r ) , ≤ r < ρ = R S ∗ SG ( T ) is the root of the equation n ( r ) = 1 /
3. For 0 < r ≤ R S ∗ SG ( T ), we have n ( r ) ≥ / e . That is, r ( − r )( − r ) ≤ e − e + 1 . For the class T , the centre of the disk is 1, therefore the disk obtained in (2.4) is containedin the region bounded by the modified sigmoid, using equation (3.7). For the function f ( z ) = z (1 + z ) e z , at z = − R S ∗ SG ( T ) = − ρ , (cid:12)(cid:12)(cid:12)(cid:12) log zf ′ ( z ) /f ( z )2 − ( zf ′ ( z ) /f ( z )) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (1 − ρ + ρ ) / (1 − ρ )2 − ((1 − ρ + ρ ) / (1 − ρ )) (cid:12)(cid:12)(cid:12)(cid:12) = 1 . ( iii ) The function defined s ( r ) = 1 − r/ √ r , ≤ r < ρ = R S ∗ SG ( T ) is the root of the equation s ( r ) = 2 / (1 + e ). For 0 < r ≤ R S ∗ SG ( T ), we have s ( r ) ≥ / e . That is 3 r √ r ≤ e − e + 1 . For the class T , the centre of the disk is 1, therefore the disk obtained in (2.6) is containedin the region bounded by the modified sigmoid, by equation (3.7). For the function f ( z ) = z ( z + √ z ) , at z = − R S ∗ SG ( T ) = − ρ , (cid:12)(cid:12)(cid:12)(cid:12) log zf ′ ( z ) /f ( z )2 − ( zf ′ ( z ) /f ( z )) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (3 ρ ) / p ρ − (1 − (3 ρ ) / ( p ρ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 . References [1] R. M. Ali, N. K. Jain and V. Ravichandran, Radii of starlikeness associated with the lemniscate ofBernoulli and the left-half plane, Appl. Math. Comput. (2012), no. 11, 6557–6565.[2] R. M. Ali, N. K. Jain and V. Ravichandran, On the radius constants for classes of analytic functions,Bull. Malays. Math. Sci. Soc. (2) (2013), no. 1, 23–38.[3] R. M. Ali, K. Sharma and V. Ravichandran, Starlikeness of analytic functions with subordinate ratios,preprint.[4] S. Afis and K. I. Noor, On subclasses of functions with boundary and radius rotations associated withcrescent domains, Bull. Korean Math. Soc. (2020), no. 6, 1529–1539.[5] N. E. Cho, V. Kumar, S. S. Kumar and V. Ravichandran, Radius problems for starlike functionsassociated with the sine function, Bull. Iranian Math. Soc. (2019), no. 1, 213–232.[6] S. Gandhi and Ravichandran, V. Starlike functions associated with a lune. Asian-Eur. J. Math. 10(2017), no. 4, 1750064, 12 pp.[7] P. Goel and S. Sivaprasad Kumar, Certain class of starlike functions associated with modified sigmoidfunction, Bull. Malays. Math. Sci. Soc. (2020), no. 1, 957–991.[8] S. Kumar and V. Ravichandran, A subclass of starlike functions associated with a rational function,Southeast Asian Bull. Math. (2016), no. 2, 199–212.[9] T. H. MacGregor, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. (1963), 514–520.[10] T. H. MacGregor, The radius of univalence of certain analytic functions. II, Proc Amer. Math. Soc. (1963), 521–524.[11] T. H. MacGregor, A class of univalent functions, Proc. Amer. Math. Soc. (1964), 311–317.[12] R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associatedwith exponential function, Bull. Malays. Math. Sci. Soc. (2015), no. 1, 365–386.[13] V. Ravichandran and K. Sharma, Sufficient conditions for starlikeness, J. Korean Math. Soc. (2015),no. 4, 727—749.[14] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer.Math. Soc. 118 (1993), no. 1, 189–196.[15] T. N. Shanmugam and V. Ravichandran, Certain properties of uniformly convex functions, Computa-tional Methods And Function Theory 1994 (Penang), 319–324, Ser. Approx. Decompos., 5 World Sci.Publ., River Edge, NJ.[16] K. Sharma, N. E. Cho and V. Ravichandran, Sufficient conditions for strong starlikeness, Bull. IranianMath. Soc. (2020), DOI:10.1007/s41980-020-00452-z.[17] K. Sharma, N. K. Jain and V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat.(Springer) (2016), no. 5, 923–939.[18] K. Sharma and V. Ravichandran, Applications of subordination theory to starlike functions, Bull.Iranian Math. Soc. (2016), no. 3, 761–777.[19] K. Sharma and V. Ravichandran, Sufficient conditions for Janowski starlike functions, Stud. Univ.Babe¸s-Bolyai Math. (2016), no. 1, 63–76.[20] L. A. Wani and A. Swaminathan, Starlike and convex functions associated with a nephroid domain,Bull. Malays. Math. Sci. Soc. (2020). https://doi.org/10.1007/s40840-020-00935-6 Department of Mathematics, National Institute of Technology, Tiruchirappalli–620015, India
Email address : [email protected] Department of Mathematics, Atma Ram Sanatan Dharma College, University of Delhi,Delhi–110 021, India
Email address : [email protected]; [email protected] Department of Mathematics, National Institute of Technology, Tiruchirappalli–620015, India
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