On Geometrical Properties of Certain Analytic functions
OON GEOMETRICAL PROPERTIES OF CERTAIN ANALYTICFUNCTIONS
S. SIVAPRASAD KUMAR AND KAMALJEET GANGANIA
Abstract.
We introduce the class of analytic functions F ( ψ ) := (cid:26) f ∈ A : (cid:18) zf (cid:48) ( z ) f ( z ) − (cid:19) ≺ ψ ( z ) , ψ (0) = 0 (cid:27) , where ψ is univalent and establish the growth theorem with some geometric con-ditions on ψ and obtain the Koebe domain with some related sharp inequalities.Note that functions in this class may not be univalent. As an application, weobtain the growth theorem for the complete range of α and β for the functionsin the classes BS ( α ) := { f ∈ A : ( zf (cid:48) ( z ) /f ( z )) − ≺ z/ (1 − αz ) , α ∈ [0 , } and S cs ( β ) := { f ∈ A : ( zf (cid:48) ( z ) /f ( z )) − ≺ z/ ((1 − z )(1 + βz )) , β ∈ [0 , } ,respectively which improves the earlier known bounds. The sharp Bohr-radii forthe classes S ( BS ( α )) and BS ( α ) are also obtained. A few examples as well ascertain newly defined classes on the basis of geometry are also discussed. . Primary 30C80, Secondary 30C45. Keywords and Phrases . Subordination, Bohr-Radius, Majorization, Distortion the-orem. 1.
Introduction
Let A denotes the class of analytic functions of the form f ( z ) = z + (cid:80) ∞ k =2 a k z k inthe open unit disk ∆ := { z : | z | < } . Let f ( z ) = w and Γ w be the image of Γ z (thecircle C r : z = re iθ ) under the function f in A . The curve Γ w is said to be starlikewith respect to w = 0 if arg( w − w ) is a non-decreasing function of θ , that is, ddθ arg( w − w ) ≥ , θ ∈ [0 , π ] , which is equivalent to ddθ arg( w − w ) = (cid:60) (cid:18) zf (cid:48) ( z ) f ( z ) (cid:19) ≥ . (1.1)If the inequality (1.1) holds for each circle | z | = r <
1, then it characterizes a specialclass S ∗ , the class of starlike functions in ∆. It is obvious from (1) that for each0 < r <
1, the curve Γ w is not allowed to have a loop which ensure the univalency ofthe function. But if for some 0 (cid:54) = z ∈ ∆, (cid:60) ( zf (cid:48) ( z ) /f ( z )) <
0, then f is not starlikewith respect to 0, or equivalently we can say that the image curve Γ w : f ( | z | = r )is definitely not starlike, but still it may or may not be univalent. From (1.1), wealso see the importance of the Carathe´odory functions by writing (1.1) in terms of The work of the second author is supported by University Grant Commission, New-Delhi, Indiaunder UGC-Ref. No.:1051/(CSIR-UGC NET JUNE 2017). a r X i v : . [ m a t h . C V ] S e p S. SIVAPRASAD KUMAR AND KAMALJEET subordination as: zf (cid:48) ( z ) f ( z ) ≺ z − z ( z ∈ ∆) , (1.2)where the symbol ≺ stands for the usual subordination. In 1992, Ma-Minda [8]generalized (1.2) by unifying all the subclasses of starlike functions as follows: S ∗ (Ψ) := (cid:26) f ∈ A : zf (cid:48) ( z ) f ( z ) ≺ Ψ( z ) (cid:27) , (1.3)where Ψ has positive real part, Ψ(∆) symmetric about the real axis with Ψ (cid:48) (0) > ψ in ∆such that ψ (0) = 0, ψ (∆) is starlike with respect to 0 and introduce the followingclass of analytic functions: F ( ψ ) := (cid:26) f ∈ A : zf (cid:48) ( z ) f ( z ) − ≺ ψ ( z ) , ψ (0) = 0 (cid:27) . (1.4)Note that when 1 + ψ ( z ) (cid:54)≺ (1 + z ) / (1 − z ), then the functions in the class F ( ψ )may not be univalent in ∆ which also implies F ( ψ ) (cid:54)⊆ S ∗ . Thus in case, when thefunction 1 + ψ := Ψ has positive real part, Ψ(∆) symmetric about the real axiswith Ψ (cid:48) (0) >
0, then F ( ψ ) reduces to the class S ∗ (Ψ). The functions in the classdefined in (1.3) are univalent which help a lot in studying the geometrical propertiesof the functions, for example, Growth and Distortion theorems etc. But this maynot be quite easy to study the analogous results in the class F ( ψ ). In this direction,recently, Kargar et al. [6] considered the following class, the first of it’s kind: BS ( α ) := (cid:26) f ∈ A : zf (cid:48) ( z ) f ( z ) − ≺ z − αz , α ∈ [0 , (cid:27) , (1.5)where z/ (1 − αz ) =: ψ ( z ) (Booth lemniscate function [11] and [12]) is an analyticunivalent function and symmetric with respect to the real and imaginary axes. Notethat the function (1 + z/ (1 − αz )) assumes negative values for α (cid:54) = 0, thereforefunctions in this class may not be univalent. For f belonging to BS ( α ), using thevertical strip domain { w ∈ C : µ < (cid:60) w < ν, where µ < < ν } , Kargar etal. [6] proved that | f ( z ) /z | is bounded and obtained the coefficient estimates when0 ≤ α ≤ − √ k − th roottransformation. In 2018, Najmadi et al. [10] obtained the bounds for the quantities (cid:60) f ( z ), | f ( z ) | and | f (cid:48) ( z ) | when 0 ≤ α ≤ − √
2. Recently, Kargar et al. [7] obtainedthe best dominant of the subordination f ( z ) /z ≺ F ( z ) for the range 0 < α ≤ − √ F ( z ) = (1 + z √ α ) / (1 − z √ α ) √ α . Cho etal. [3] dealt with the first order differential subordination implications and also solvedthe various sharp radius problems pertaining to the class BS ( α ).In 2019, Masih et al. [9] considered the following class with β ∈ [0 , / S cs ( β ) := (cid:26) f ∈ A : (cid:18) zf (cid:48) ( z ) f ( z ) − (cid:19) ≺ z (1 − z )(1 + βz ) , β ∈ [0 , (cid:27) . (1.6)They proved the growth theorem and also obtained the sharp estimates for thelogarithmic coefficients but only for the range β ∈ [0 , / β ∈ [0 , / S cs ( β ) is a Ma-Minda subclass, but for the other range, functions in this class maynot be univalent. N GEOMETRICAL PROPERTIES OF CERTAIN ANALYTIC FUNCTIONS 3
In this paper, we establish the sharp growth theorem for the class F ( ψ ) withcertain geometric conditions on ψ and obtain covering theorem. Further providesome examples including newly defined classes are also discussed. As an application,we obtain growth theorem for the complete range of α and β for the functions inthe classes BS ( α ) and S cs ( β ), respectively that improves the earlier known bounds.Finally, the sharp Bohr-radii for the classes S ( BS ( α )) and BS ( α ) are obtained.For some classes, we study the geometrical behavior of an analytic function of theform f ( z ) /z which arises frequently while working with the class S ∗ (Ψ) and play animportant role, for example, in obtaining the bounds for (cid:60) ( f ( z ) /z ) and arg( f ( z ) /z ).The geometric properties and coefficients estimation for the class F ( ψ ) are still open.2. Main Results
Let F ( ψ ) be the class as defined in (1.4). Now we begin with the following: Theorem 2.1 (Growth Theorem) . If max | z | = r (cid:60) ψ ( z ) = ψ ( r ) and min | z | = r (cid:60) ψ ( z ) = ψ ( − r ) . Then f ∈ F ( ψ ) satisfies the sharp inequalities: r exp (cid:18)(cid:90) r ψ ( − t ) t dt (cid:19) ≤ | f ( z ) | ≤ r exp (cid:18)(cid:90) r ψ ( t ) t dt (cid:19) , ( | z | = r ) . (2.1) Proof.
Let f ∈ F ( ψ ). For z = re iθ , we have φ ( − r ) ≤ (cid:60) ψ ( re iθ ) ≤ φ ( r ) . (2.2)Let Φ( z ) = ψ ( ω ( z )), where ω is a Schwarz function. Then from (1.3), we havelog f ( z ) z = (cid:90) z Φ( ζ ) ζ dζ. Now by taking ζ = te iβ so that dζ = e iβ dt , where β is fixed but arbitrary and z = re iβ , we have log f ( z ) z = (cid:90) r Φ( te iβ ) t dt. (2.3)From the Maximum-minimum modulus principle, we find that Φ also satisfies theinequality (2.2). Therefore, without loss of generality, we may replace Φ by ψ and β by θ in (2.3). Then by equating real parts on either side of (2.3), we havelog (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) z (cid:12)(cid:12)(cid:12)(cid:12) = (cid:90) r (cid:60) Φ( te iθ ) t dt (2.4)and now using the inequalities (2.2) in (2.4), we obtain (cid:90) r ψ ( − t ) t dt ≤ log (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) z (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) r ψ ( t ) t dt, and (2.1) follows. The result is sharp for the function f ( z ) = z exp (cid:90) z ψ ( t ) t dt. (2.5)This completes the proof. Remark . In the above theorem, we chose max | z | = r (cid:60) ψ ( z ) = ψ ( r ) and min | z | = r (cid:60) ψ ( z ) = ψ ( − r ) for computational convenience. However, these conditions may change ac-cording to the choice of ψ in that case, appropriately these may be replaced. S. SIVAPRASAD KUMAR AND KAMALJEET
Remark . If 1+ ψ is a Carath´eodory univalent function, then Theorem 2.1 reducesto the result [8, Corollary 1, p. 161] and moreover, letting r tends to 1 in Theorem 2.1,we obtain the covering theorem (Koebe-radius) for the class F ( ψ ). Corollary 2.2 (Covering Theorem) . If f ∈ F ( ψ ) and f as defined in (2.5) , theneither f is a rotation of f or { w ∈ ∆ : | w | ≤ − f ( − } ⊂ f (∆) , where − f ( −
1) = lim r → ( − f ( − r )) . Let L ( f, r ) denotes the length of the boundary curve f ( | z | = r ). Note that for z = re iθ , we have L ( f, r ) := (cid:82) π | zf (cid:48) ( z ) | dθ . Now we obtain the following result: Corollary 2.3.
Assume that max | z | = r | ψ ( z ) | = ψ ( r ) and also ψ satisfies the condi-tions of Theorem 2.1. Let M ( r ) = exp (cid:16)(cid:82) r ψ ( t ) t dt (cid:17) . If f ∈ F ( ψ ) , then for | z | = r ,we have (cid:60) f ( z ) z ≤ M ( r ) , | f (cid:48) ( z ) | ≤ (1 + ψ ( r )) M ( r ) and L ( f, r ) ≤ πr (1 + ψ ( r )) M ( r ) . Let ψ ( z ) = (cid:26) βz/ (1 + αz ) , β >
0, 0 < α < ηz, η > ψ are clearly convex univalent and ψ (∆) are symmetricabout real axis as ψ ( z ) = ψ (¯ z ). It is further evident that 1 + ψ ( z ) (cid:54)≺ (1 + z ) / (1 − z )except for the second choice of ψ when 0 < η ≤
1. We now obtain the followingsharp result from Theorem 2.1:
Example 2.4.
Let f ∈ F ( βz/ (1 + αz )) , where β > and < α < and | z | = r .Then r (1 − αr ) βα ≤ | f ( z ) | ≤ r (1 + αr ) βα , which implies: (cid:110) w : | w | ≤ (1 − α ) βα (cid:111) ⊂ f (∆) , | f (cid:48) ( z ) | ≤ (cid:18) βr αr (cid:19) (1+ αr ) βα and (cid:60) f ( z ) z ≤ (1+ αr ) βα . Example 2.5.
Let f ∈ F ( ηz ) , where η > and | z | = r . Then r exp( − ηr ) ≤ | f ( z ) | ≤ r exp( ηr ) , which implies: { w : | w | ≤ exp( − η ) } ⊂ f (∆) , | f (cid:48) ( z ) | ≤ (1 + ηr ) exp( ηr ) and (cid:60) f ( z ) z ≤ exp( ηr ) . From the above examples, it is clear that f ∈ F ( ψ ) if and only if zf (cid:48) ( z ) f ( z ) ∈ (cid:26) Ω , when ψ ( z ) = βz/ (1 + αz );Ω , when ψ ( z ) = ηz, where Ω = { w ∈ C : | w − | < | β − α ( w − |} and Ω = { w ∈ C : | w − | < η } ,respectively for z ∈ ∆. N GEOMETRICAL PROPERTIES OF CERTAIN ANALYTIC FUNCTIONS 5 Some Applications and Further results
On Booth-Lemniscate.
Let BS ( α ) be the class as defined in (1.5). Theorem 3.1.
Let < α < and f ∈ BS ( α ) , then for | z | = r − ˆ f ( − r ) ≤ | f ( z ) | ≤ ˆ f ( r ) , (3.1) where ˆ f ( z ) = z (cid:18) z √ α − z √ α (cid:19) √ α . (3.2) The result is sharp.
Proof.
Let ψ ( z ) := z/ (1 − αz ) and f ∈ BS ( α ) := F ( ψ ). For z = re iθ , we have − r − αr ≤ (cid:60) ψ ( re iθ ) ≤ r − αr and − (cid:90) r − αt dt ≤ log (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) z (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) r − αt dt, where (cid:90) r − αt dt = 12 √ α log 1 + √ αr − √ αr . Hence, the result follows from Theorem 1.4.
Remark . Theorem 3.1 improves the upper bound of (cid:60) f ( z ) and bounds of | f ( z ) | ,obtained in [10, Theorem 2, p. 116] and [10, Theorem 3, p. 116] respectively.We now extend [7, Theorem 2.6, p. 1238] for the complete range of α usingTheorem 3.1: Corollary 3.2.
Let f ∈ BS ( α ) , α ∈ (0 , and | z | = r , then (cid:60) f ( z ) z ≤ (cid:18) r √ α − r √ α (cid:19) √ α and | f (cid:48) ( z ) | ≤ (cid:18) r − αr (cid:19) (cid:18) r √ α − r √ α (cid:19) √ α . The result is sharp for the function ˆ f given in (3.2) . Corollary 3.3.
Let α ∈ (0 , be fixed. Then f ∈ BS ( α ) satisfies the inequality L ( f, r ) ≤ πr (cid:18) r − αr (cid:19) (cid:18) r √ α − r √ α (cid:19) √ α , ( | z | = r ) . Corollary 3.4 (Koebe-radius) . Let < α < and ˆ f as given in (3.2) . If f ∈BS ( α ) , then either f is a rotation of ˆ f or { w ∈ C : | w | ≤ − ˆ f ( − } ⊂ f (∆) . Proof.
The proof follows by letting r tends to 1 in the inequality − ˆ f ( − r ) ≤ | f ( z ) | ,given in (3.1). Theorem 3.5.
Let α ∈ (0 , − √ be fixed. Then f ∈ BS ( α ) satisfies the sharpinequality (cid:12)(cid:12)(cid:12)(cid:12) arg f ( z ) z (cid:12)(cid:12)(cid:12)(cid:12) ≤ max | z | = r arg (cid:18) z √ α − z √ α (cid:19) √ α . S. SIVAPRASAD KUMAR AND KAMALJEET
Proof.
From [7, Theorem 2.5, p. 1238], we have f ( z ) /z ≺ ˆ f ( z ) /z for 0 < α ≤ −√ f is defined in (3.2). Since the function ˆ f ( z ) /z is convex and symmetric aboutthe real axis in ∆, therefore we easily see that (cid:18) − √ α √ α (cid:19) √ α > . Thus ˆ f ( z ) /z is a Carathe´odory function and the result follows.For our next result, we need the following definition and a related class: Definition 3.6.
Let f ( z ) = (cid:80) ∞ k =0 a k z k and g ( z ) = (cid:80) ∞ k =0 b k z k are analytic in ∆ and f (∆) = Ω. Consider a class of analytic functions S ( f ) := { g : g ≺ f } or equivalently S (Ω) := { g : g ( z ) ∈ Ω } . Then the class S ( f ) is said to satisfy Bohr-phenomenon, ifthere exists a constant r ∈ (0 ,
1] such that the inequality (cid:80) ∞ k =1 | b k | r k ≤ d ( f (0) , ∂ Ω)holds for all | z | = r ≤ r , where d ( f (0) , ∂ Ω) denotes the Euclidean distance between f (0) and the boundary of Ω = f (∆). The largest such r for which the inequalityholds, is called the Bohr-radius.See the articles [1, 2] and the references therein for more. Let us now introducethe following class: S ( BS ( α )) := (cid:26) g : g ≺ f, g ( z ) = ∞ (cid:88) k =1 b k z k and f ∈ BS ( α ) (cid:27) . Theorem 3.7 (Booth-Bohr-radius) . The class S ( BS ( α )) satisfies Bohr-phenomenonin | z | ≤ r ( α ) , where r ( α ) is the unique positive root of the equation r (cid:18) r √ α − r √ α (cid:19) √ α − (cid:18) − √ α √ α (cid:19) √ α = 0 , (3.3) whenever < α ≤ − √ . The result is sharp for the function ˆ f given in (3.2) . Proof.
Since g ∈ S ( BS ( α )), we have g ≺ f for a fixed f ∈ BS ( α ). From Corol-lary 3.4, we obtain the Koebe-radius r ∗ = − ˆ f ( −
1) such that r ∗ ≤ d (0 , ∂ Ω) = | f ( z ) | for | z | = 1. Also using [7, Theorem 2.5, p. 1238], we have f ( z ) z ≺ ˆ f ( z ) z . (3.4)Recall the result [2, Lemma 1, p.1090], which reads as: let f and g be analytic in∆ with g ≺ f, where f ( z ) = (cid:80) ∞ n =0 a n z n and g ( z ) = (cid:80) ∞ k =0 b k z k . Then (cid:80) ∞ k =0 | b k | r k ≤ (cid:80) ∞ n =0 | a n | r n for | z | = r ≤ / . Now using the result for g ≺ f and (3.4), we have ∞ (cid:88) k =1 | b k | r k ≤ r + ∞ (cid:88) n =2 | a n | r n ≤ ˆ f ( r ) for | z | = r ≤ / . Finally, to establish the inequality (cid:80) ∞ k =1 | b k | r k ≤ d ( f (0) , ∂ Ω) , it is enough to showˆ f ( r ) ≤ r ∗ . But this holds whenever r ≤ r ( α ), where r ( α ) is the least positive rootof the equation ˆ f ( r ) = r ∗ . Now let T ( r ) := ˆ f ( r ) − r ∗ , then T (cid:48) ( r ) = (cid:18) r √ α − r √ α (cid:19) √ α + r (cid:18) r √ α − r √ α (cid:19) √ α − − r √ α ) . Since (1 + r √ α ) / (1 − r √ α ) >
0, therefore T (cid:48) ( r ) > T is an increasingfunction of r . Also T (0) < T (1) >
0. Thus the existence of the root r ( α ) N GEOMETRICAL PROPERTIES OF CERTAIN ANALYTIC FUNCTIONS 7 is ensured by the Intermediate Value theorem for the continuous functions. By acomputation, it can easily be seen that r ( α ) < / Corollary 3.8.
Let < α ≤ − √ . The Bohr-radius for the class BS ( α ) is r ( α ) ,where r ( α ) is the unique positive root of the Eq. (3.3) . On Cissoid of Diocles.
Let us consider S β ( z ) = z (1 − z )(1 + βz ) = 11 + β (cid:18) − z + 11 + βz (cid:19) = ∞ (cid:88) n =1 − ( − β ) n β z n , where β ∈ [0 , Cissoid of Diocles : CS ( β ) := (cid:26) w = u + iv ∈ C : (cid:18) u − β − (cid:19) ( u + v ) + 2 β (1 + β ) ( β − v = 0 (cid:27) . Let us now consider the class S cs ( β ) as defined in (1.6). Masih et al. [9] consideredthis class with β ∈ [0 , /
2] since (cid:60) (1+ z/ ((1 − z )(1 + βz )) ≥ (2 β − / (2( β − ≥ S cs ( β ) = F ( S β ( z )) for β ∈ [0 ,
1) and we have the following result:
Theorem 3.9.
Let f ∈ S cs ( β ) and β ∈ [0 , . Then − ˜ f ( − r ) ≤ | f ( z ) | ≤ ˜ f ( r ) , where ˜ f ( z ) = z (cid:18) βz − z (cid:19) β . (3.5) Proof.
Let ψ ( z ) := z/ ((1 − z )(1 + βz )) and f ∈ S ∗ cs ( β ) := F ( ψ ). Following theproof of [9, Theorem 3.1, p. 5], it is easy to see that for z = re iθ , where θ ∈ [0 , π ],we have min | z | = r (cid:60) ψ ( z ) = − r + ( β − r + βr (1 + r ) (1 − βr ) = ψ ( − r )and max | z | = r (cid:60) ψ ( z ) = lim θ → − r + βr − βr cos θ + r cos θ (1 + r − r cos θ )(1 + β r + 2 βr cos θ ) ≤ β − β ) = max | z | =1 (cid:60) ψ ( z ) . Thus, we have ψ ( − r ) ≤ (cid:60) ψ ( z ) ≤ ψ ( r ) for r (cid:54) = 1 and 1 / (2( β − ψ ( − ≤(cid:60) ψ ( z ) ≤ ( β − / (2( β + 1) ) for r = 1. Also note that˜ f ( z ) = z exp (cid:90) z ψ ( t ) t dt = z (cid:18) βz − z (cid:19) β . Now the result follows from Theorem 1.4.
Remark . Let ˜ F ( z ) = ˜ f ( z ) /z and | z | = 1 , where ˜ f is as defined in Theorem 3.9.A calculation show that1 + ˜ F (cid:48)(cid:48) ( z )˜ F (cid:48) ( z ) = 1 + − βz (1 + βz )(1 − z ) + 2 z − z , which implies that (cid:60) (cid:32) F (cid:48)(cid:48) ( z )˜ F (cid:48) ( z ) (cid:33) ≥ β (cid:60) (cid:18) − z (1 + βz )(1 − z ) (cid:19) . S. SIVAPRASAD KUMAR AND KAMALJEET
Since (cid:60) (cid:18) − z (1 + βz )(1 − z ) (cid:19) = 1 − β − β + 2 β cos θ ) =: g ( θ ) , and a simple calculation shows that g attains its minimum at θ = 0. Therefore, wehave 1 + ˜ F (cid:48)(cid:48) ( z )˜ F (cid:48) ( z ) ≥ β (1 − β )2(1 + β ) ≥ . Hence ˜ F is convex univalent in ∆.Observe that the function S β ( z ) is not convex when β (cid:54) = 0 and the result, f ( z ) /z ≺ ˜ F ( z ) similar to theorem 3.14 is still open for f ∈ S cs ( β ). By letting r tends to 1 inthe above Theorem 3.9, we obtain: Corollary 3.10 (Koebe-radius) . Let ˜ f as given in (3.5) . If f ∈ S cs ( β ) , then either f is a rotation of ˜ f or (cid:40) w ∈ C : | w | ≤ − ˜ f ( −
1) = (cid:18) − β (cid:19) / (1+ β ) (cid:41) ⊂ f (∆) . Remark . We improved the result [9, Corollary 4.3.1, p. 8] in Theorem 3.9 andCorollary 3.10 by extending the range of β .3.3. Modified Koebe function:
The Koebe function k ( z ) = z/ (1 − z ) has a poleat z = 1 and maps unit disk onto the domain C − ( −∞ , / K ( z ) := z (1 + ηz ) , ≤ η < , (3.6)which is bounded in ∆ and symmetric about the real-axis. It is interesting to observethe geometry of the domain K (∆), which assumes different shapes for differentchoices of η such as a convex or a Bean or a Cardioid shaped domain. Especiallywhen η tends to 1, we see that one of the rotation of the image domain K (∆) willconverge to k (∆) and thereby justifying the name of K ( z ). Since k ( z ) = ( u ( z ) − /
4, where u ( z ) = (1 + z ) / (1 − z ), in a similar fashion, we can write K ( z ) = 14 η (1 − v ( z )) , where v ( z ) = (1 − ηz ) / (1 + ηz ) and η (cid:54) = 0 . Lemma 3.1.
The function K ( z ) as defined in (3.6) is convex for ≤ η ≤ − √ . Proof.
Let K ( z ) = z/ (1 + ηz ) . When η = 0, K ( z ) is the identity function andhence is convex. So let us consider 0 < η <
1. By a computation, we obtain that1 + zK (cid:48)(cid:48) ( z ) K (cid:48) ( z ) = 1 − ηz + η z (1 − ηz )(1 + ηz ) . Putting z = e iθ , we have (cid:60) (cid:18) zK (cid:48)(cid:48) ( z ) K (cid:48) ( z ) (cid:19) = 1 − η (1 − η ) cos θ − η (1 + cos θ )sin θ ((1 + η ) − (2 η cos θ ) ) . (3.7)Since ((1 + η ) − (2 η cos θ ) ) > θ and for each fixed η . Therefore, we nowonly need to consider the numerator in (3.7). A computation reveals that N ( θ ) := 1 − η (1 − η ) cos θ − η (1 + cos θ )sin θ N GEOMETRICAL PROPERTIES OF CERTAIN ANALYTIC FUNCTIONS 9 is increasing in 0 ≤ θ ≤ π (note that N ( θ ) = N ( − θ )) with N ( θ ) ≥ < η ≤ − √
3, while N ( θ ) takes negative values when η > − √
3. Hence by the definitionof convexity, result follows.Now let us consider the function ψ ( z ) := γz (1 + ηz ) = γK ( z ) , where γ > , and introduce a related class defined as follows: S γ ( η ) := (cid:26) f ∈ A : (cid:18) zf (cid:48) ( z ) f ( z ) − (cid:19) ≺ γz (1 + ηz ) , η ∈ [0 , , γ > (cid:27) . (3.8)Note that if γ and η satisfies the condition (1 − η ) ≥ γ , then the class S γ ( η )reduces to a Ma-Minda subclass of univalent starlike functions. Also letting η = 1 / γ = 25( √ − /
16, we see that the class S ∗ ( √ z ) ⊂ S γ ( η ). Theorem 3.11.
Let f ∈ S γ ( η ) and η ∈ [0 , − √ . Then − κ ( − r ) ≤ | f ( z ) | ≤ κ ( r ) , where κ ( z ) := z exp (cid:18) γz (1 + ηz ) (cid:19) . Proof.
Since ψ ( z ) = γK ( z ), using Lemma 3.1, we see that for | z | = r , ψ ( − r ) ≤(cid:60) ψ ( z ) ≤ ψ ( r ) . Also, we have κ ( z ) = z exp (cid:82) z ( ψ ( t ) /t ) dt . Hence, the result followsfrom Theorem 1.4.Using Lemma 3.1, we also obtain that (cid:60) ψ ( z ) ≥ ψ ( − r ) for all η ∈ [0 ,
1) whichimplies − κ ( − r ) ≤ | f ( z ) | . So we have the following results: Corollary 3.12 (Radius of starlikeness) . Let f ∈ S γ ( η ) , γ > and η ∈ [0 , . Then f is starlike (univalent) of order α ∈ [0 , inside the disk | z | < r , where r is thesmallest positive root of the equation (1 − α ) η r − (2(1 − α ) η + γ ) r + (1 − α ) = 0 . Corollary 3.13 (Koebe-radius) . Let f ∈ S γ ( η ) and η ∈ [0 , . Then either f is arotation of κ or (cid:26) w ∈ C : | w | ≤ − κ ( −
1) = exp (cid:18) − γ (1 − η ) (cid:19)(cid:27) ⊂ f (∆) . Remark . Let F κ ( z ) := κ ( z ) /z = exp( γz/ (1 + ηz ) ). We see that for η = 0 and γ ≤ F κ is clearly convex. So consider 0 < η <
1. After some calculations, weobtain that G ( z ) := 1 + zF (cid:48)(cid:48) κ ( z ) F (cid:48) κ ( z ) = η z + (2 η + γη ) z − (6 η + 2 ηγ ) z + ( γ − η ) z + 1(1 + ηz ) (1 − ηz ) . Now for z = e iθ , the denominator of the real part of G is (1 + η − η cos θ )(1 + η +2 η cos θ ) >
0, since (1 − η ) > G is nonnegative if and only if 0 < γ < < η ≤ η , where η (depends on γ ) is thesmallest positive root of the equation(1 − γ ) + (3 γ − η + 12 η + (8 − γ ) η − η + (2 + γ ) η + 4 η − η = 0 . (3.9)Hence, F κ convex for 0 < γ < < η ≤ η . For our next result, we need to recall the following result of Ruscheweyh andStankiewicz [13]:
Lemma 3.2 ([13]) . Let the analytic functions F and G be convex univalent in ∆ .If f ≺ F and g ≺ G , then f ∗ g ≺ F ∗ G ( z ∈ ∆) . Theorem 3.14.
Let η ∈ [0 , − √ . If f belongs to the class S γ ( η ) , then f ( z ) z ≺ F κ ( z ) , ( z ∈ ∆) where F κ ( z ) = κ ( z ) /z is the best dominant and κ as defined in Theorem 3.11. Proof.
Let f ∈ S γ ( η ), then φ ( z ) := zf (cid:48) ( z ) f ( z ) − ≺ ψ ( z ) . (3.10)It is well-known that the function g ( z ) = log (cid:18) − z (cid:19) = ∞ (cid:88) n =1 z n n ∈ C , where C is the usual class of normalized convex(univalent) function and thus for f ∈ A , we get φ ( z ) ∗ g ( z ) = (cid:90) z φ ( t ) t dt and ψ ( z ) ∗ g ( z ) = (cid:90) z ψ ( t ) t dt. (3.11)From Lemma 3.1, we see that ψ is convex for η ∈ [0 , − √ φ ( z ) ∗ g ( z ) ≺ ψ ( z ) ∗ g ( z ) . (3.12)Now from (3.11) and (3.12), we obtain (cid:90) z φ ( t ) t dt ≺ (cid:90) z ψ ( t ) t dt, which implies that f ( z ) z := exp (cid:90) z φ ( t ) t dt ≺ exp (cid:90) z ψ ( t ) t dt =: κ ( z ) z . This completes the proof.
Corollary 3.15.
Let < γ < and < η ≤ min { − √ , η } , where η is theleast positive root of the equation (3.9) and also let < γ ≤ π/ when η = 0 . If f ∈ S γ ( η ) , then f satisfies the sharp inequality (cid:12)(cid:12)(cid:12)(cid:12) arg f ( z ) z (cid:12)(cid:12)(cid:12)(cid:12) ≤ max | z | = r arg exp (cid:18) γz (1 + ηz ) (cid:19) . Proof.
Let F κ ( z ) := κ ( z ) /z = exp( γz/ (1 + ηz ) ) which is symmetric about the realaxis. From Theorem 3.14, have f ( z ) /z ≺ F κ ( z ) for 0 ≤ η ≤ − √
3. Since for η = 0, (cid:60) F κ ( z ) > γ ≤ π/
2. The result is obvious. Now from Remark 3.4, we
N GEOMETRICAL PROPERTIES OF CERTAIN ANALYTIC FUNCTIONS 11 see that if 0 < γ < < η ≤ min { − √ , η } , where η is the least positiveroot of the equation (3.9) then F κ is convex which implies (cid:60) F κ ( z ) ≥ exp (cid:18) − γ (1 − η ) (cid:19) > , and F κ is also a Carathe´odory function in this case. Hence the result follows.Now using Theorem 3.11, Remark 3.4 and Theorem 3.14, we obtain the followingresult: Theorem 3.16.
Let f ∈ S γ ( η ) , then (cid:60) (cid:18) f ( z ) z (cid:19) ≤ exp (cid:18) γr (1 + ηr ) (cid:19) for η ∈ [0 , and min | z | = r exp (cid:18) γz (1 + ηz ) (cid:19) ≤ (cid:60) (cid:18) f ( z ) z (cid:19) for η ∈ [0 , − √ . In partcular, if < γ < and < η ≤ min { − √ , η } , where η is the least positiveroot of the equation (3.9) , then exp (cid:18) − γr (1 − ηr ) (cid:19) ≤ (cid:60) (cid:18) f ( z ) z (cid:19) . The result is sharp.
We conclude this paper by introducing the following three new subclasses of F ( ψ ): T := (cid:26) f ∈ A : zf (cid:48) ( z ) f ( z ) − ≺ log(1 − z ) (cid:27) , which means zf (cid:48) ( z ) /f ( z ) ∈ { w ∈ C : | exp( w − − | < } , S p := (cid:26) f ∈ A : zf (cid:48) ( z ) f ( z ) ≺ − (cid:18) log 1 + √ z − √ z (cid:19) (cid:27) , or equivalently zf (cid:48) ( z ) /f ( z ) ∈ { w ∈ C : | − w | < (cid:60) ((1 − w ) + π ) } , a parabola withopening in left half plane and L ( β ) := (cid:26) f ∈ A : zf (cid:48) ( z ) f ( z ) − ≺ z cos( βz ) , β ∈ [0 , (cid:27) . The above new classes are still open to study. Also see figure 1. Note that for theclasses T and L ( β ), the function f defined in (2.5) takes the respective particularform f T ( z ) := z exp( − Li ( z )) , where − (cid:90) z log(1 − t ) t dt = ∞ (cid:88) n =1 z n n =: Li ( z )known as dilogarithm function and f L ( z ) := z exp (cid:90) z βt dt = z (sec βz + tan βz ) /β ) , β (cid:54) = 0 . Γ = z / cos z Γ = log ( - z ) Γ Γ - - - - - Figure 1.
Boundary curves of the functions z/ cos z and log(1 − z ) Conclusion
It is interesting to observe that even in the class F ( ψ ), functions may not beunivalent. But with the conditions on the bounds for the real part of ψ , a similarresult holds as obtained by Ma-Minda [8] which is quiet important to obtain theKoebe domain. From Remark 3.2 and Remark 3.4, we also note that the function f ( z ) /z , where f as defined in (2.5) behaves quite differently in the particularclasses. Conflict of interest
The authors declare that they have no conflict of interest.
References [1] ALI, R. M.—JAIN, N. K.—RAVICHANDRAN, V.:
Bohr radius for classes of analytic func-tions , Results Math. (2019), Art. 179, 13 pp.[2] BHOWMIK, B.—DAS, N.: Bohr phenomenon for subordinating families of certain univalentfunctions , J. Math. Anal. Appl. (2018), 1087–1098.[3] CHO, N. E.—KUMAR, S.—KUMAR, V. —RAVICHANDRAN, V.:
Differential subordinationand radius estimates for starlike functions associated with the Booth lemniscate , Turkish J.Math. (2018), 1380–1399.[4] GOEL, P.—SIVAPRASAD KUMAR, S.: Certain class of starlike functions associated withmodified sigmoid function , Bull. Malays. Math. Sci. Soc. (2020), 957–991.[5] KANAS, S.—WI´SNIOWSKA, A.: Conic domains and starlike functions , Rev. RoumaineMath. Pures Appl. (2000), 647–657.[6] KARGAR, R.—EBADIAN, A.—SOK ´O(cid:32)L, J.: On Booth lemniscate and starlike functions ,Anal. Math. Phys. (2019), 143–154.[7] KARGAR, R.—EBADIAN, A.—TROJNAR-SPELINA, L.: Further results for starlike func-tions related with Booth lemniscate , Iran. J. Sci. Technol. Trans. A Sci. (2019), 1235–1238.[8] MA, W.C.—MINDA,D.: A unified treatment of some special classes of univalent functions , Proceedings of the Conference on Complex Analysis (Tianjin, 1992) , Conf. Proc. Lecture NotesAnal., I Int. Press, Cambridge, MA., 157–169.[9] MASIH, V. S.—EBADIAN, A.—YALC¸ IN, S.:
Some properties associated to a certain classof starlike functions , Math. Slovaca (2019), 1329–1340.[10] NAJMADI, P.—NAJAFZADEH, SH.—EBADIAN, A.: Some properties of analytic functionsrelated with Booth lemniscate , Acta Univ. Sapientiae Math. (2018), 112–124. N GEOMETRICAL PROPERTIES OF CERTAIN ANALYTIC FUNCTIONS 13 [11] PIEJKO, K.—SOK ´O(cid:32)L, J.:
Hadamard product of analytic functions and some special regionsand curves , J. Inequal. Appl. , (2013), Art. 420, 13 pp.[12] PIEJKO, K.—SOK ´O(cid:32)L, J.:
On Booth lemniscate and hadamard product of analytic functions ,Math. Slovaca. (2015), 1337–1344.[13] RUSCHEWEYH, S.—STANKIEWICZ, J.: Subordination under convex univalent functions ,Bull. Polish Acad. Sci. Math. (1985), 499–502.[14] SHARMA, P.—RAINA, R. K.—SOK ´O(cid:32)L, J.: Certain Ma-Minda type classes of analytic func-tions associated with the crescent-shaped region , Anal. Math. Phys. (2019), 1887–1903. Department of Applied Mathematics, Delhi Technological University, Delhi–110042, India
E-mail address : [email protected] Department of Applied Mathematics, Delhi Technological University, Delhi–110042, India
E-mail address ::