Geometric Properties of Generalized Bessel Function associated with the Exponential Function
GGEOMETRIC PROPERTIES OF GENERALIZED BESSEL FUNCTIONASSOCIATED WITH THE EXPONENTIAL FUNCTION
ADIBA NAZ, SUMIT NAGPAL, AND V. RAVICHANDRAN
Dedicated to Prof. Ajay Kumar on the occasion of his retirement
Abstract.
Sufficient conditions are determined on the parameters such that the gener-alized and normalized Bessel function of the first kind and other related functions belongto subclasses of starlike and convex functions defined in the unit disk associated withthe exponential mapping. Several differential subordination implications are derived foranalytic functions involving Bessel function and the operator introduced by Baricz et al. [Differential subordinations involving generalized Bessel functions, Bull. Malays. Math.Sci. Soc. (2015), no. 3, 1255–1280]. These results are obtained by constructingsuitable class of admissible functions. Examples involving trigonometric and hyperbolicfunctions are provided to illustrate the obtained results. Introduction
Due to diverse applications of Bessel function in wave propagation and static poten-tial, it is widely studied in geometric function theory. Consider the generalized Besseldifferential equation z ω (cid:48)(cid:48) ( z ) + bzω (cid:48) ( z ) + ( cz − ν + ν (1 − b )) ω ( z ) = 0 ( z, ν, b, c ∈ C ) . (1.1)The particular solution of (1.1) is known as the generalized Bessel function ω ν,b,c of thefirst kind of order ν having the infinite series representation ω ν,b,c ( z ) = (cid:88) n ≥ ( − c ) n n !Γ( ν + n + ( b + 1) / (cid:16) z (cid:17) n + ν ( z, ν, b, c ∈ C ) (1.2)where Γ denotes the Euler gamma function. Set J ν := ω ν, , and I ν := ω ν, , − . Then J ν and I ν are the Bessel function of the first kind of order ν and the modified Besselfunction of the first kind of order ν respectively. In fact, they are the solutions of theBessel’s equation z ω (cid:48)(cid:48) ( z ) + zω (cid:48) ( z ) + ( z − ν ) ω ( z ) = 0 and modified Bessel equation z ω (cid:48)(cid:48) ( z ) + zω (cid:48) ( z ) − ( z + ν ) ω ( z ) = 0 having the form J ν ( z ) = (cid:88) n ≥ ( − n n !Γ( ν + n + 1) (cid:16) z (cid:17) n + ν ( z, ν ∈ C ) (1.3)and I ν ( z ) = (cid:88) n ≥ n !Γ( ν + n + 1) (cid:16) z (cid:17) n + ν ( z, ν ∈ C ) (1.4) Mathematics Subject Classification.
Key words and phrases.
Bessel function, modified Bessel function, starlike functions, convex functions,differential subordination, exponential function. a r X i v : . [ m a t h . C V ] J a n ADIBA NAZ, SUMIT NAGPAL, AND V. RAVICHANDRAN respectively. Similarly, the functions j ν := √ πω ν, , / i ν := √ πω ν, , − / ν and the modified spherical Besselfunction of the first kind of order ν having the form j ν ( z ) = (cid:114) π z J ν +1 / ( z ) = √ π (cid:88) n ≥ ( − n n !Γ( ν + n + 3 / (cid:16) z (cid:17) n + ν ( z, ν ∈ C ) (1.5)and i ν ( z ) = (cid:114) π z I ν +1 / ( z ) = √ π (cid:88) n ≥ n !Γ( ν + n + 3 / (cid:16) z (cid:17) n + ν ( z, ν ∈ C ) (1.6)respectively. Therefore the study of the geometric properties of ω ν,b,c such as univalence,starlikeness and convexity in the unit disk gives a unified treatment to the study ofBessel, modified Bessel, spherical Bessel and modified spherical Bessel functions. Formore information on Bessel functions, see [1–5, 7, 10, 14, 16–18] and references therein.Let A be the class of all analytic functions f in D := { z ∈ C : | z | < } normalized bythe condition f (0) = 0 = f (cid:48) (0) − S be its subclass consisting of univalent functions.The notion of subordination is quite useful in describing the containment relationshipbetween the image domains of two analytic functions. For any two analytic functions f and g defined in D , we say that f is subordinate to g , written as f ≺ g , if there existsa Schwarz function s with s (0) = 0 and | s ( z ) | < f ( z ) = g ( s ( z )) for all z ∈ D . It is easy to see that if f ≺ g , then f (0) = g (0) and f ( D ) ⊆ g ( D ). In particular, if g is univalent in D , then f ≺ g if and only if f (0) = g (0) and f ( D ) ⊆ g ( D ). Consider thefamily P e consisting of functions p that are analytic in D with p (0) = 1 and p ( z ) ≺ e z .Since Re( e z ) > z ∈ D , a function p ∈ P e also satisfies Re( p ( z )) > D andtherefore p is also a Carath´eodory function.Since the normalization plays a vital role in studying the properties of univalent func-tions, therefore, it is imperative to normalize the generalized Bessel function ω ν,b,c givenby (1.1). By means of the transformation ϕ ν,b,c ( z ) = 2 ν Γ (cid:18) ν + b + 12 (cid:19) z − ν/ ω ν,b,c ( √ z ) (1.7)and using the Pochhammer symbol ( x ) µ given by( x ) µ = Γ( x + µ )Γ( x ) = (cid:40) , µ = 0 x ( x + 1) · · · ( x + n − , µ = n ∈ N the function ϕ ν,b,c has the infinite series representation ϕ ν,b,c ( z ) = (cid:88) n ≥ ( − c/ n ( κ ) n z n n ! (1.8)where ν , b , c ∈ C such that κ = ν +( b +1) / / ∈ { , − , − , . . . } and is called the generalizedand normalized Bessel function of the first kind of order ν .Note that zϕ ν,b,c ∈ A . If we set b n := ( − c/ n / ( n !( κ ) n ), then the function ϕ ν,b,c / ∈ A asit is not normalized but ( ϕ ν,b,c ( z ) − b ) /b ∈ A . For brevity, we shall denote ϕ ν,b,c by ϕ ν and zϕ ν,b,c is denoted by ϑ ν,b,c or simply by ϑ ν . Also, note that the function ϕ ν is entire EOMETRIC PROPERTIES OF GENERALIZED BESSEL FUNCTION 3 as the radius of convergence of the series (1.8) is infinity and it also satisfies the secondorder differential equation 4 z ϕ (cid:48)(cid:48) ν ( z ) + 4 κzϕ (cid:48) ν ( z ) + czϕ ν ( z ) = 0 . (1.9)Moreover, it satisfies the recurrence relation [2, Lemma 1.2, p. 14]4 κϕ (cid:48) ν ( z ) = − cϕ ν +1 ( z ) . (1.10)Determining the sufficient conditions on the parameters ν , b , c of generalized and nor-malized Bessel functions to belong to well-known classes of univalent functions have along history. Baricz and Ponnusamy [4] determined the conditions on the parametersof normalized form of generalized Bessel functions to be convex and starlike in D . Us-ing the theory of differential subordination, Bohra and Ravichandran [6] determined theconditions so that ϕ ν,b,c is strongly convex of order 1 /
2. Recently, Madaan et al. [11]determined the lemniscate convexity and other properties of the generalized and normal-ized Bessel function using the theory of differential subordination. Kanas et al. [10] andMondal and Dhuain [14] independently obtained the conditions on the parameters forwhich ϕ ν,b,c is Janowski convex and zϕ (cid:48) ν,b,c is Janowski starlike in the unit disk. In Section2, we determine the conditions on the parameters κ and c so that the generalized andnormalized Bessel function belongs to the classes K e and S *e introduced and studied byMendiratta et al. [12]. Here K e and S *e are the subclasses of S consisting of functions f for which the quantity w = 1 + zf (cid:48)(cid:48) /f (cid:48) (and w = zf (cid:48) /f respectively) lies in the imagedomain | log w | < w = e z . In terms of subordination, f ∈ S *e if and only if zf (cid:48) /f ∈ P e and f ∈ K e if and only if 1 + zf (cid:48)(cid:48) /f (cid:48) ∈ P e . Interestingexamples involving trigonometric and hyperbolic functions are provided to illustrate ourresults and its connection with Libera operator is also established.Using the technique of convolution to define a linear operator involving generalizedhypergeometric functions by Dziok and Srivastava [9], recently Baricz et al. [3] introducedthe B cκ -operator defined by B cκ f ( z ) := ( ϑ ν ∗ f )( z ) = z + ∞ (cid:88) n =1 ( − c/ n a n +1 ( κ ) n z n +1 n ! ( z ∈ D )where f ( z ) = z + (cid:80) ∞ n =1 a n +1 z n +1 ∈ A . Note that B cκ -operator satisfies the recurrencerelation z ( B cκ +1 f ( z )) (cid:48) = κB cκ f ( z ) − ( κ − B cκ +1 f ( z ) ( z ∈ D ) . (1.11)Applying the methodology of admissible functions given by Miller and Mocanu [13], Baricz et al. [3] proved various differential subordination results involving the B cκ -operator. Mo-tivated by their work, we derive a class of admissible functions involving the B cκ -operatorin Section 3. Sufficient conditions are also determined so that the function B cκ f belongsto the classes K e and S *e .We will make use of the theory of differential subordination which was introducedby Miller and Mocanu [13] in proving the results of Sections 2 and 3. Let Q denotethe set of functions q that are analytic and univalent on D \ E ( q ) where E ( q ) = { ζ ∈ ∂ D : lim z → ζ q ( z ) = ∞} and q (cid:48) ( ζ ) (cid:54) = 0 for ζ ∈ ∂ D \ E ( q ). Further, we denote the subclassof Q for which q (0) = a by Q a . The following definition of admissible functions andfoundation result of differential subordination theory will be needed in our study. ADIBA NAZ, SUMIT NAGPAL, AND V. RAVICHANDRAN
Definition . [13, Definition 2.3(a), p. 27] Let Ω be any subset in C and q ∈ Q . DefineΨ(Ω , q ) to be a class of admissible functions ψ : C × D → C that satisfies the admissibilitycondition ψ ( r, s, t ; z ) / ∈ Ω whenever r = q ( ζ ), s = mζq (cid:48) ( ζ ) and Re(1 + t/s ) ≥ m Re(1 + ζq (cid:48)(cid:48) ( ζ ) /q (cid:48) ( ζ )) where z ∈ D , ζ ∈ ∂ D \ E ( q ) and m ≥ Theorem 1.2. [13, Theorem 2.3(b), p. 28] Let ψ ∈ Ψ(Ω , q ) with q ∈ Q a . If an analyticfunction p ( z ) = a + a n z n + a n +1 z n +1 + · · · satisfies ψ ( p ( z ) , zp (cid:48) ( z ) , z p (cid:48)(cid:48) ( z ); z ) ∈ Ω , then p ( z ) ≺ q ( z ) . Exponential Convexity and Starlikeness of the Generalized andNormalized Bessel Function
Using the theory of differential subordination, Naz et al. [15] investigated the class ofadmissible functions associated with the exponential function e z and proved the followinglemma. Lemma 2.1. [15] Let Ω be a subset of C and the function ψ : C × D → C satisfies theadmissibility condition ψ ( r, s, t ; z ) / ∈ Ω whenever r = e e iθ , s = me iθ r and Re(1 + t/s ) ≥ m (1 + cos θ ) where z ∈ D , θ ∈ [0 , π ) and m ≥ . If p is an analytic function in D with p (0) = 1 and ψ ( p ( z ) , zp (cid:48) ( z ) , z p (cid:48)(cid:48) ( z ); z ) ∈ Ω for z ∈ D , then the function p ∈ P e . Observe that the admissibility condition ψ ( r, s, t ; z ) (cid:54)∈ Ω is true for all r = e e iθ , s = me iθ e e iθ and t such that Re(1 + t/s ) ≥
0, that is, Re(( t + s ) e − iθ e − e iθ ) ≥ θ ∈ [0 , π )and m ≥
1. We ought to employ this form of admissibility condition to demonstrate ourresults. Furthermore, for the case ψ : C × D → C , the admissibility condition reduces to ψ ( e e iθ , me iθ e e iθ ; z ) / ∈ Ωwhere z ∈ D , θ ∈ [0 , π ) and m ≥ κ and c so that the generalized andnormalized Bessel function belongs to the class P e using the recurrence relation (1.10). Theorem 2.2.
If the parameters κ , c ∈ C are constrained such that Re( κ ) ≥ ( | c | /
4) + 1 ,then the function ϕ ν ∈ P e .Proof. If c = 0, then ϕ ν ≡ P e . Suppose that c (cid:54) = 0. In order toprove our result, we first claim that the function ( − κ/c ) ϕ (cid:48) ν ∈ P e if Re( κ ) ≥ | c | /
4. Letus define a function p : D → C by p ( z ) = − κc ϕ (cid:48) ν ( z ) . Then p is analytic in D and p (0) = 1. Suppose that z (cid:54) = 0. Since ϕ ν satisfies the differentialequation (1.9), we have4 zϕ (cid:48)(cid:48) ν ( z ) + 4 κϕ (cid:48) ν ( z ) + cϕ ν ( z ) = 0 ( z ∈ D ) . Differentiation gives 4 zϕ (cid:48)(cid:48)(cid:48) ν ( z ) + 4( κ + 1) ϕ (cid:48)(cid:48) ν ( z ) + cϕ (cid:48) ν ( z ) = 0 so that the function p satisfies4 z p (cid:48)(cid:48) ( z ) + 4( κ + 1) zp (cid:48) ( z ) + czp ( z ) = 0 ( z ∈ D ) . (2.1) EOMETRIC PROPERTIES OF GENERALIZED BESSEL FUNCTION 5
Note that the above equation is also valid for z = 0. Suppose Re( κ ) ≥ | c | / ψ : C × D → C by ψ ( r, s, t ; z ) = 4 t + 4( κ + 1) s + czr and let Ω := { } , then(2.1) can be written as ψ ( p ( z ) , zp (cid:48) ( z ) , z p (cid:48)(cid:48) ( z ); z ) ∈ Ω ( z ∈ D ) . To prove the required result, we will make use of Lemma 2.1 which will be applicable ifwe show that ψ ( r, s, t ; z ) / ∈ Ω whenever r = e e iθ , s = me iθ e e iθ and Re(( s + t ) e − iθ e − e iθ ) ≥ z ∈ D , θ ∈ [0 , π ) and m ≥
1. For any z , z ∈ C , recall the inequality | z + z | ≥|| z | − | z || and note that | ψ ( r, s, t ; z ) | = | t + s ) + 4 κme iθ e e iθ + cze e iθ | > e cos θ (cid:18) | ( t + s ) e − iθ e − e iθ + κm | − | c | (cid:19) ≥ e cos θ (cid:18) Re (cid:0) ( t + s ) e − iθ e − e iθ (cid:1) + m Re( κ ) − | c | (cid:19) ≥ . Thus | ψ ( r, s, t ; z ) | (cid:54) = 0 and using Lemma 2.1, we conclude that p ( z ) ≺ e z for all z ∈ D , thatis, if c (cid:54) = 0 and Re( κ ) ≥ | c | /
4, then the function ( − κ/c ) ϕ (cid:48) ν ∈ P e . Using the recurrencerelation (1.10), we have 4( κ − ϕ (cid:48) ν − ( z ) = − cϕ ν ( z ) . Hence it follows that if c (cid:54) = 0 and Re( κ ) ≥ ( | c | /
4) + 1, then the function ϕ ν ∈ P e . Thiscompletes the proof of the theorem. (cid:3) It is interesting to note that generalized Bessel functions of first kind can be reducedto elementary trigonometric and hyperbolic functions. Therefore the above deduced re-sult is significant and leads to various interesting relations involving the generalized andnormalized Bessel function and trigonometric functions by selecting the suitable choicesof the parameters involved. Let us illustrate Theorem 2.2 by the following example.
Example . Clearly, the following functions ϕ ( z ) = ϕ , , ( z ) = sin( √ z ) √ zϕ ( z ) = ϕ , , ( z ) = 112 (cid:32) √ √ z ) z / − √ z ) z (cid:33) ϕ ( z ) = ϕ , , ( z ) = 140 (cid:32) − z ) cos( √ z ) z + 63(1 − z ) sin( √ z ) √ z / (cid:33) satisfy the hypothesis of Theorem 2.2. Therefore ϕ i ( z ) ≺ e z (for i = 1 , , ϕ ( z ) = ϕ , , ( z ) ≺ e z and ϕ / ( z ) = ϕ / , , ( z ) ≺ e z can be interpreted graphically as shown in Figure 2.The next result deals with a sufficient condition on the parameters κ and c so that thegeneralized and normalized Bessel function belongs to the class K e . ADIBA NAZ, SUMIT NAGPAL, AND V. RAVICHANDRAN j ( ) D exp( ) D - - (a) c = 2 , κ = 3 / exp( ) D - - j ( ) D (b) c = 6 , κ = 5 / j ( ) D exp( ) D - - (c) c = 10 , κ = 9 / Figure 1.
Graph showing ϕ i ( D ) ⊂ exp( D ) ( i = 1 , , exp( ) D - - (a) c = 30 , κ = 17 / exp( ) D - - (b) c = 60 , κ = 16 Figure 2.
Graph showing ϕ i ( D ) ⊂ exp( D ) ( i = 8 , / Theorem 2.4.
Let the parameters κ , c ∈ C be constrained such that c (cid:54) = 0 , Re( κ ) ≥ | c | / and | κ − | + 14( e − | c | ≤ e + e − e ( e −
1) (2.2) then the function − κ ( ϕ ν − /c ∈ K e .Proof. Since Re( κ ) ≥ | c | / c (cid:54) = 0, ϕ ν is univalent in D by means of [2, Theorem 2.3,p. 29] and therefore ϕ (cid:48) ν ( z ) (cid:54) = 0 for all z ∈ D . Define a function p : D → C by p ( z ) = 1 + z ( − κ ( ϕ ν − /c ) (cid:48)(cid:48) ( − κ ( ϕ ν − /c ) (cid:48) = 1 + zϕ (cid:48)(cid:48) ν ( z ) ϕ (cid:48) ν ( z ) . Then p is analytic in D with p (0) = 1. For z (cid:54) = 0, the differential equation (1.9) yields4 zϕ (cid:48)(cid:48)(cid:48) ν ( z ) + 4( κ + 1) ϕ (cid:48)(cid:48) ν ( z ) + cϕ (cid:48) ν ( z ) = 0. As ϕ (cid:48) ν ( z ) (cid:54) = 0 for all z ∈ D , dividing the previousequation by ϕ (cid:48) ν ( z ) and then multiplying it by z , we obtain4 (cid:18) z ϕ (cid:48)(cid:48)(cid:48) ν ( z ) ϕ (cid:48) ν ( z ) (cid:19) + 4( κ + 1) (cid:18) zϕ (cid:48)(cid:48) ν ( z ) ϕ (cid:48) ν ( z ) (cid:19) + cz = 0 . EOMETRIC PROPERTIES OF GENERALIZED BESSEL FUNCTION 7
By making use of the equation zϕ (cid:48)(cid:48) ν ( z ) /ϕ (cid:48) ν ( z ) = p ( z ) −
1, a straightforward calculationshows that z ϕ (cid:48)(cid:48)(cid:48) ν ( z ) ϕ (cid:48) ν ( z ) = zp (cid:48) ( z ) + p ( z ) − p ( z ) + 2 . Hence the function p satisfies the differential equation4 zp (cid:48) ( z ) + 4 p ( z ) + 4( κ − p ( z ) −
1) + cz − z = 0. Define a function ψ ( r, s ; z ) := 4 s +4 r +4( κ − r − cz − { } . Then ψ ( p ( z ) , zp (cid:48) ( z ); z ) ∈ Ω for all z ∈ D . Again, we will employ Lemma2.1 to prove that p ( z ) ≺ e z for all z ∈ D , that is, we will show that ψ ( r, s ; z ) / ∈ Ω where r = e e iθ and s = me iθ e e iθ for z ∈ D , θ ∈ [0 , π ) and m ≥ | ψ ( r, s ; z ) | = 4 (cid:12)(cid:12)(cid:12)(cid:12) s + r − κ − r −
1) + 14 cz (cid:12)(cid:12)(cid:12)(cid:12) > (cid:18) | s + r − | − | κ − || r − | − | c | (cid:19) . For r = e e iθ , s = me iθ e e iθ , θ ∈ [0 , π ), z ∈ D and m ≥
1, we get | s + r − | = ( me cos θ cos( θ + sin θ ) + e θ cos(2 sin θ ) − + ( me cos θ sin( θ + sin θ ) + e θ sin(2 sin θ )) := g ( θ ) . Using the second derivative test, the function g attains its minimum value at θ = π sothat min θ ∈ [0 , π ) g ( θ ) = g ( π ) = (cid:18) − me + 1 e − (cid:19) which implies | s + r − | ≥ me − e + 1 ≥ e − e + 1 . Similarly, we have | r − | = ( e cos θ cos(sin θ ) − + e θ sin (sin θ )= 1 + e θ − e cos θ cos(sin θ ) := g ( θ ) . The function g attains its maximum value at θ = 0 and thus | r − | ≤ e −
1. Using thesecalculations, we obtain | ψ ( r, s ; z ) | = | ψ ( e e iθ , me iθ e e iθ ; z ) | > (cid:18) e − e + 1 − ( e − | κ − | − | c | (cid:19) ≥ | ψ ( r, s ; z ) | > (cid:3) Note that if | κ − | + 14( e − | c | ≤ e + e − e ( e −
1) (2.3)then the function − κ − ϕ ν − − /c ∈ K e by Theorem 2.4 provided that c (cid:54) = 0 andRe( κ ) ≥ ( | c | /
4) + 1. Using the famous Alexander duality theorem between the classes K e and S *e , that is, f ∈ K e if and only if zf (cid:48) ∈ S *e , we obtain − κ − zϕ (cid:48) ν − /c ∈ S *e . Alsothe recurrence relation (1.10) gives czϕ ν ( z ) = − κ − zϕ (cid:48) ν − ( z ). Consequently, we havethe following result. ADIBA NAZ, SUMIT NAGPAL, AND V. RAVICHANDRAN
Theorem 2.5.
Let the parameters κ , c ∈ C be constrained such that c (cid:54) = 0 , Re( κ ) ≥ ( | c | /
4) + 1 and (2.3) is satisfied, then the function ϑ ν ∈ S *e where ϑ ν ( z ) = zϕ ν ( z ) for all z ∈ D . The particular choices of b and c in Theorems 2.4 and 2.5 leads to the correspondingresults for Bessel ( b = c = 1), modified Bessel ( b = 1 and c = − b = 2and c = 1) and modified spherical Bessel ( b = 2 and c = −
1) functions. For Bessel andmodified Bessel function, we have | c | = 1 and κ = ν + 1 so that the following corollary isobtained. Corollary 2.6.
Let the parameter ν ∈ C . For the functions J ν ( z ) := 2 ν Γ( ν + 1) z − ν/ J ν ( √ z ) and I ν ( z ) := 2 ν Γ( ν + 1) z − ν/ I ν ( √ z ) where J ν and I ν are the Bessel function and the modified Bessel function of the first kindof order ν defined by (1.3) and (1.4) respectively, the following assertions hold.(a) If Re( ν ) ≥ − . and | ν − | ≤ e + 34( e − then the functions − ν + 1)( J ν − ∈ K e and − ν + 1)( I ν − ∈ K e .(b) If Re( ν ) ≥ . and | ν − | ≤ e + 34( e − then the functions z J ν ∈ S *e and z I ν ∈ S *e . To illustrate Corollary 2.6, consider the functions J / ( z ) = (cid:114) π z − / J / ( √ z ) = sin √ z √ z and I / ( z ) = (cid:114) π z − / I / ( √ z ) = sinh √ z √ z . These functions satisfy the hypothesis of Corollary 2.6(a). Therefore both the functions − J / −
1) and − I / −
1) belong to the class K e . In terms of subordination, it canbe written as 1 + z J (cid:48)(cid:48) / ( z ) J (cid:48) / ( z ) = (1 − z ) sin √ z − √ z cos √ z √ z cos √ z − √ z ≺ e z and 1 + z I (cid:48)(cid:48) / ( z ) I (cid:48) / ( z ) = (1 + z ) sinh √ z − √ z cosh √ z √ z cosh √ z − √ z ≺ e z . Similarly, the functions J / ( z ) = 3 (cid:114) π z − / J / ( √ z ) and I / ( z ) = 3 (cid:114) π z − / I / ( √ z )satisfy Corollary 2.6(b) and therefore the functions z J / ( z ) = 3 (cid:18) sin √ z √ z − cos √ z (cid:19) and z I / ( z ) = 3 (cid:18) cosh √ z − sinh √ z √ z (cid:19) EOMETRIC PROPERTIES OF GENERALIZED BESSEL FUNCTION 9 are in S ∗ e . Also, the similar reasoning shows that the functions z J / ( z ) = 15((3 − z ) sin √ z − √ z cos √ z ) z / and z I / ( z ) = 15((3 + z ) sinh √ z − √ z cos √ z ) z / belong to the class S ∗ e .In the similar fashion, if we take | c | = 1 and κ = ν + (3 / Corollary 2.7.
Let the parameter ν ∈ C . For the functions j ν ( z ) := 1 √ π ν +1 Γ (cid:18) ν + 32 (cid:19) z − ν/ j ν ( √ z ) and i ν ( z ) := 1 √ π ν +1 Γ (cid:18) ν + 32 (cid:19) z − ν/ i ν ( √ z ) where j ν and i ν are the spherical Bessel function and the modified spherical Bessel functionof the first kind of order ν defined by (1.5) and (1.6) respectively, the following assertionshold.(a) If Re( ν ) ≥ − . and | ν − | ≤ e + 32( e − then the functions − ν + 3)( j ν − ∈ K e and − ν + 3)( i ν − ∈ K e .(b) If Re( ν ) ≥ − . and | ν − | ≤ e + 32( e − then the functions z j ν ∈ S *e and z i ν ∈ S *e . It has been proved in [12] that both K e and S *e are closed under convolution with theconvex functions. We will make use of this observation to study the behaviour of anintegral operator. The Libera operator L : A → A is defined as L [ f ]( z ) := 2 z (cid:90) z f ( t ) dt = − z + log(1 − z )) z ∗ f ( z )where f ∈ A and z ∈ D . Theorems 2.4 and 2.5 yield the following result which helps inconstructing starlike and convex functions involving ϕ ν and ϑ ν . Corollary 2.8.
If the parameters ν , b , c are constrained as in Theorem 2.4, then thefunction ( − κ ( ϕ ν − /c ) ∗ f ∈ K e for every convex function f ∈ A and in particular, L [ − κ ( ϕ ν − /c ] belongs to the class K e . Similarly, if the parameters ν , b , c are con-strained as in Theorem 2.5, then the function ϑ ν ∗ f ∈ S *e for every convex function f ∈ A and in particular, L [ ϑ ν ] belongs to the class S *e . Let us make use of this corollary to obtain new functions in the classes K e and S *e . Bythe discussion succeeding Corollary 2.6, it follows that L [ − J / − z ) = 12 z ( z + 2 cos √ z − L [ − I / − z ) = 12 z ( z − √ z + 2)belong to K e . Similarly, it can be deduced that L [ z J / ]( z ) = − z ( √ z sin √ z + 2 cos √ z − L [ z I / ]( z ) = 12 z ( √ z sinh √ z − √ z + 2)are in the class S *e .The last theorem of this section gives a sufficient condition under which the function2 ν Γ( κ ) z − ν ω ν,b,c ∈ S *e where ω ν,b,c is given by (1.2). To prove this, we shall be requiringthe following result by Miller and Mocanu [13, Theorem 2.3h, p. 34] for M = 1 / a = 0. Lemma 2.9.
Let Ω be a subset of C and the function ψ : C × D → C satisfies theadmissibility condition ψ ( r, s, t ; z ) / ∈ Ω whenever r = e iθ / , s = me iθ / and Re(1 + t/s ) ≥ m where z ∈ D , θ ∈ [0 , π ) and m ≥ . If p is an analytic function in D with p (0) = 0 and ψ ( p ( z ) , zp (cid:48) ( z ) , z p (cid:48)(cid:48) ( z ); z ) ∈ Ω for z ∈ D , then | p ( z ) | < / . As noticed earlier, the admissibility condition ψ ( r, s, t ; z ) / ∈ Ω in Lemma 2.9 is verifiedfor all r = e iθ / s = me iθ / t + s ) e − iθ ) ≥ / θ ∈ [0 , π ) and m ≥ Theorem 2.10.
If the parameters κ ∈ R , c ∈ C are constrained such that κ ≥ max (cid:26) | c | + 1 , | c | + 34 (cid:27) (2.4) then the function ν Γ( κ ) z − ν ω ν,b,c ∈ S *e .Proof. As κ ≥ | c | / ϕ ν ( z ) (cid:54) = 0 for all z ∈ D by Theorem 2.2. Therefore the function p : D → C defined by p ( z ) = zϕ (cid:48) ν ( z ) ϕ ν ( z )is analytic in D with p (0) = 0. Firstly, we claim that | p ( z ) | < / z ∈ D .Since p ( z ) ϕ ν ( z ) = zϕ (cid:48) ν ( z ) and the function ϕ ν satisfies (1.9), it follows that (4 zp (cid:48) ( z ) +4 p ( z ) + 4( κ − p ( z ) + cz ) ϕ ν ( z ) = 0 for all z ∈ D . Set q ( z ) := 4 zp (cid:48) ( z ) + 4 p ( z ) + 4( κ − p ( z ) + cz , then the preceding equation can be rewritten as q ( z ) ϕ ν ( z ) = 0 for all z ∈ D .Differentiating this equation and then multiplying it by z gives (cid:0) zq (cid:48) ( z ) + q ( z ) p ( z ) (cid:1) ϕ ν ( z ) =0 for all z ∈ D . As ϕ ν ( z ) (cid:54) = 0 for all z ∈ D , we have zq (cid:48) ( z ) + q ( z ) p ( z ) = 0 for all z ∈ D .Hence the function p satisfies the differential equation4 (cid:0) z p (cid:48)(cid:48) ( z ) + κzp (cid:48) ( z ) + 3 zp ( z ) p (cid:48) ( z ) + p ( z ) + ( κ − p ( z ) (cid:1) + ( p ( z ) + 1) cz = 0 . Rearrangement of the terms yields4 (cid:0) z p (cid:48)(cid:48) ( z ) + zp (cid:48) ( z ) + ( κ − zp (cid:48) ( z ) + p ( z )) + 3 zp ( z ) p (cid:48) ( z ) + p ( z ) (cid:1) + ( p ( z ) + 1) cz = 0 . (2.5)Define a function ψ ( r, s, t ; z ) : C × D → C by ψ ( r, s, t ; z ) = 4 (cid:0) t + s + ( κ − s + r ) + 3 sr + r (cid:1) + ( r + 1) cz and suppose that Ω := { } . Then (2.5) can be written as ψ ( p ( z ) , zp (cid:48) ( z ) , z p (cid:48)(cid:48) ( z ); z ) ∈ Ω.To prove our claim, we shall use Lemma 2.9 which will be apt if we prove that ψ ( r, s, t ; z ) / ∈ EOMETRIC PROPERTIES OF GENERALIZED BESSEL FUNCTION 11
Ω whenever r = e iθ / s = me iθ / t + s ) e − iθ ) ≥ / θ ∈ [0 , π ) and m ≥ | ψ ( r, s, t ; z ) | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) t + s + ( κ − (cid:18) me iθ e iθ (cid:19) + 316 me iθ + e iθ (cid:19) + (cid:18) e iθ (cid:19) cz (cid:12)(cid:12)(cid:12)(cid:12) > (cid:12)(cid:12)(cid:12)(cid:12) t + s ) e − iθ + ( κ − (cid:18) m + e iθ (cid:19) + 34 me iθ (cid:12)(cid:12)(cid:12)(cid:12) − − (cid:12)(cid:12)(cid:12)(cid:12) e iθ (cid:12)(cid:12)(cid:12)(cid:12) · | c |≥ (cid:0) ( t + s ) e − iθ (cid:1) + ( κ − (cid:18) m + cos θ (cid:19) + 34 m cos θ − − (cid:12)(cid:12)(cid:12)(cid:12) e iθ (cid:12)(cid:12)(cid:12)(cid:12) · | c |≥ κ − − | c | ≥ . Thus | ψ ( r, s, t ; z ) | (cid:54) = 0 and using Lemma 2.9, we conclude that | p ( z ) | < / z ∈ D ,that is (cid:12)(cid:12)(cid:12)(cid:12) zϕ (cid:48) ν ( z ) ϕ ν ( z ) (cid:12)(cid:12)(cid:12)(cid:12) <
14 ( z ∈ D ) . Now let us suppose h ( z ) := 2 ν Γ( κ ) z − ν ω ν,b,c ( z ). Using (1.7), it follows that h ( z ) = zϕ ν ( z ). We need to show that h ∈ S *e , that is, | log zh (cid:48) ( z ) /h ( z ) | < z ∈ D . Bymaking use of the fact that if | z | < /
2, then | log(1 + z ) | ≤ | z | / (cid:12)(cid:12)(cid:12)(cid:12) log zh (cid:48) ( z ) h ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) z ϕ (cid:48) ν ( z ) ϕ ν ( z ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) z ϕ (cid:48) ν ( z ) ϕ ν ( z ) (cid:12)(cid:12)(cid:12)(cid:12) < < . This completes the proof of the theorem. (cid:3)
As a particular case of Theorem 2.10, the functions 2 ν Γ( ν + 1) z − ν J ν and 2 ν Γ( ν +1) z − ν I ν belongs to S *e if ν ≥ / ≈ . J ν and I ν are the Bessel functionand the modified Bessel function of the first kind of order ν defined by (1.3) and (1.4)respectively. For instance, the functions2 / Γ(5 / z − / J / = 3(sin z − z cos z ) z and 2 / Γ(5 / z − / I / = 3( z cosh z − sinh z ) z are in the class S *e .3. Exponential Starlikeness Associated with the B cκ -operator In this section, we will be investigating the differential subordination results involvingthe B cκ -operator and its connection with exponential function. Throughout this section,we shall consider the function ϑ ν = zϕ ν where ϕ ν is given by (1.8). Theorem 3.1.
Let a function f ∈ A with Re κ ≥ max { , | c | / κ ) / / } . (a) If f is a convex function and B cκ − f ∈ S *e , then B cκ f ∈ S *e . (b) If zf (cid:48) is a convex function and B cκ − f ∈ K e , then B cκ f ∈ K e . Proof.
Note that ϑ ν is a starlike function by [2, Theorem 2.12(b), p. 46]. For the proof ofpart (a), since convolution of a starlike function and a convex function is starlike, therefore B cκ f ( z ) (cid:54) = 0 for all 0 (cid:54) = z ∈ D . Let us define a function p : D → C by p ( z ) = z ( B cκ f ( z )) (cid:48) B cκ f ( z ) . Then p is analytic in D with p (0) = 1. Using (1.11), we have B cκ − f ( z ) B cκ f ( z ) = 1 κ − p ( z ) + κ − z ( B cκ − f ( z )) (cid:48) B cκ − f ( z ) = p ( z ) + zp (cid:48) ( z ) p ( z ) + κ − . Since B cκ − f ∈ S *e , we have p ( z ) + zp (cid:48) ( z ) p ( z ) + κ − ≺ e z . As Re( e z ) > z ∈ D and Re κ ≥
2, we must have Re( e z + κ − > z ∈ D .Since e z is convex univalent in D , we have p ( z ) ≺ e z using [13, Theorem 3.2(a), p. 81].Hence it follows that B cκ f ∈ S *e .For the proof of part (b), let B cκ − f ∈ K e . Alexander duality theorem between theclasses K e and S *e gives z ( B cκ − f ) (cid:48) = B cκ − ( zf (cid:48) ) ∈ S *e . By part (a), B cκ ( zf (cid:48) ) = z ( B cκ f ) (cid:48) ∈ S *e as zf (cid:48) is a convex function. By again applying Alexander duality theorem, it follows that B cκ f ∈ K e . (cid:3) Let f ( z ) = z/ (1 − z ) ∈ A . Then f is a convex function and B cκ f = ϑ ν . If J ν and I ν arethe generalized and normalized forms of Bessel function and modified Bessel function ofthe first kind of order ν respectively as defined in Corollary 2.6, then Theorem 3.1 takesthe following form. Corollary 3.2.
Let ν ∈ R with ν ≥ . If z J ν ∈ S *e , then z J ν +1 ∈ S *e . Similarly, if z I ν ∈ S *e , then z I ν +1 ∈ S *e . Since z J / and z I / are members of S *e (see the discussion succeeding Corollary 2.6),therefore Corollary 3.2 shows that z J (3 / n ∈ S *e and z I (3 / n ∈ S *e for all n ∈ N .Now, we will define a class Φ C (Ω , q ) of admissible functions and use it to prove thecorresponding theorem for differential subordination, as done by Miller and Mocanu [13]for proving Theorem 1.2 using Definition 1.1. Throughout the discussion, it is assumedthat the terms appearing in the denominator are non-zero so that all the expressions arewell-defined. Definition . Let Ω be any subset of C and q ∈ Q . The class of admissible functionsΦ C (Ω , q ) consists of those functions φ : C × D → C that satisfies the following admissiblitycondition φ ( u, v, w ; z ) / ∈ Ωwhenever u = q ( ζ ) and v = q ( ζ ) + mζq (cid:48) ( ζ ) q ( ζ ) EOMETRIC PROPERTIES OF GENERALIZED BESSEL FUNCTION 13 and Re (cid:18) − u + 3 uv − v + (1 − v ) wu − v (cid:19) ≥ m Re (cid:18) ζq (cid:48)(cid:48) ( ζ ) q (cid:48) ( ζ ) (cid:19) where z ∈ D , ζ ∈ ∂ D \ E ( q ) and m ≥ Theorem 3.4.
Let φ ∈ Φ C (Ω , q ) . If a function f ∈ A satisfies the following inclusionrelation (cid:26) φ (cid:18) z ( B cκ f ( z )) (cid:48) B cκ f ( z ) , z ( B cκ f ( z )) (cid:48)(cid:48) ( B cκ f ( z )) (cid:48) , z ( B cκ f ( z )) (cid:48)(cid:48)(cid:48) ( B cκ f ( z )) (cid:48)(cid:48) ; z (cid:19) : z ∈ D (cid:27) ⊂ Ω (3.1) then z ( B cκ f ( z )) (cid:48) B cκ f ( z ) ≺ q ( z ) ( z ∈ D ) . Proof.
Define an analytic function p : D → C by p ( z ) = z ( B cκ f ( z )) (cid:48) /B cκ f ( z ). A straight-forward calculation gives 1 + z ( B cκ f ( z )) (cid:48)(cid:48) ( B cκ f ( z )) (cid:48) = p ( z ) + zp (cid:48) ( z ) p ( z )and 1 + z ( B cκ f ( z )) (cid:48)(cid:48)(cid:48) ( B cκ f ( z )) (cid:48)(cid:48) = z p (cid:48)(cid:48) ( z ) + 3 zp ( z ) p (cid:48) ( z ) − zp (cid:48) ( z ) + p ( z ) − p ( z ) + p ( z ) zp (cid:48) ( z ) + p ( z ) − p ( z ) . Let us define the transformation from C to C by u := r, v := r + sr and w := t + 3 sr − s + r − r + rs + r − r . If we write ψ ( r, s, t ; z ) = φ ( u, v, w ; z ) = φ (cid:18) r, r + sr , t + 3 sr − s + r − r + rs + r − r ; z (cid:19) then we have ψ ( p ( z ) , zp (cid:48) ( z ) , z p (cid:48)(cid:48) ( z ); z )= φ (cid:18) z ( B cκ f ( z )) (cid:48) B cκ f ( z ) , z ( B cκ f ( z )) (cid:48)(cid:48) ( B cκ f ( z )) (cid:48) , z ( B cκ f ( z )) (cid:48)(cid:48)(cid:48) ( B cκ f ( z )) (cid:48)(cid:48) ; z (cid:19) . Therefore (3.1) can be rewritten as ψ ( p ( z ) , zp (cid:48) ( z ) , z p (cid:48)(cid:48) ( z ); z ) ∈ Ω for all z ∈ D . Alsoobserve that 1 + ts = 1 − u + 3 uv − v + (1 − v ) wu − v . Hence the admissibility condition for φ ∈ Φ C (Ω , q ) is equivalent to the admissibilitycondition for ψ as given in Definition 1.1. Evidently Theorem 1.2 gives that p ( z ) ≺ q ( z )and this completes the proof. (cid:3) If Ω (cid:54) = C is a simply connected domain, then (3.1) can be written in terms of subor-dination so that Theorem 3.4 can be reformulated and extended to the case when thebehaviour of the function q on the boundary of D is not known and the best dominantsof the resulting differential subordination can be determined. The statements and proofof these results are similar to [13, Theorem 2.3d, p. 30] and [13, Theorem 2.3e, p. 31],therefore the details are omitted. The class of admissible functions Φ C (Ω , q ) with q ( z ) = e z is denoted by Φ C (Ω , e z ) whichis characterized by the following theorem in two-dimension. Theorem 3.5.
Let Ω be any set in C and suppose that the function φ : C × D → C satisfies the admissibility condition φ ( e e iθ , e e iθ + me iθ ; z ) / ∈ Ω where z ∈ D , θ ∈ [0 , π ) and m ≥ is a positive integer. If f ∈ A satisfies the relation (cid:26) φ (cid:18) z ( B cκ f ( z )) (cid:48) B cκ f ( z ) , z ( B cκ f ( z )) (cid:48)(cid:48) ( B cκ f ( z )) (cid:48) ; z (cid:19) : z ∈ D (cid:27) ⊂ Ω then B cκ f ∈ S *e . As an application, we provide the following two examples that illustrate Theorem 3.5.This, in turn, give sufficient conditions in the form of differential inequalities for a function B cκ f to be in S *e . Example . Define a function φ ( u, v ; z ) := (1 − α ) u + αv where α > /e and let h : D → C be defined as h ( z ) = 1 + ( αe − e z. Then Ω = h ( D ) = { w ∈ C : | w − | < ( αe − /e } . We want to show that φ ∈ Φ C (Ω , e z ).For z ∈ D , θ ∈ [0 , π ) and m ≥
1, consider | φ ( e e iθ , e e iθ + me iθ ; z ) − | = (cid:12)(cid:12)(cid:12) e e iθ + αme iθ − (cid:12)(cid:12)(cid:12) = ( e cos θ cos(sin θ ) + αm cos θ − + ( e cos θ sin(sin θ ) + αm sin θ ) =: (cid:96) ( θ ) . The function (cid:96) attains its minimum value at θ = π so that | φ ( e e iθ , e e iθ + me iθ ; z ) − | ≥ αm − e + 1 ≥ α − e + 1 > α − e . Therefore the admissibility condition given in Theorem 3.5 is satisfied. Hence we concludethat if f ∈ A , α > /e and (cid:12)(cid:12)(cid:12)(cid:12) (1 − α ) z ( B cκ f ( z )) (cid:48) B cκ f ( z ) + α (cid:18) z ( B cκ f ( z )) (cid:48)(cid:48) ( B cκ f ( z )) (cid:48) (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) < α − e then B cκ f ∈ S *e . In the particular case, if f ( z ) = z/ (1 − z ) and α = 1, the followingimplication holds. (cid:12)(cid:12)(cid:12)(cid:12) zϑ (cid:48)(cid:48) ν ( z ) ϑ (cid:48) ν ( z ) (cid:12)(cid:12)(cid:12)(cid:12) < − e = ⇒ (cid:12)(cid:12)(cid:12)(cid:12) log zϑ (cid:48) ν ( z ) ϑ ν ( z ) (cid:12)(cid:12)(cid:12)(cid:12) < z ∈ D ) . By Corollary 2.6, it is not possible to prove that ϑ − / , , = z J − / and ϑ − / , , − = z I − / are in S *e . However, the preceding implication can be used to prove the same. To see this,note that the function zϑ (cid:48)(cid:48)− / , , ( z ) ϑ (cid:48)− / , , ( z ) = z ((4 + z ) cos √ z + 4 √ z sin √ z )2((6 − z ) cos √ z + √ z (6 + z ) sin √ z ) EOMETRIC PROPERTIES OF GENERALIZED BESSEL FUNCTION 15 lies inside the circle | w | = 1 − /e (see Figure 3). Thus it follows that ϑ − / , , ( z ) = z cos √ z − z cos √ z + z √ z sin √ z ∈ S *e . Similarly, it can be shown that ϑ − / , , − ( z ) = z cosh √ z + 13 z cosh √ z − z √ z sinh √ z ∈ S *e . e | | = 1 1 w - - - - - - Figure 3.
Image of zϑ (cid:48)(cid:48)− / , , ( z ) /ϑ (cid:48)− / , , ( z ) under D . Example . Let φ ( u, v ; z ) := uv . For z ∈ D , θ ∈ [0 , π ) and m ≥
1, consider | φ ( e e iθ , e e iθ + me iθ ; z ) − | = (cid:12)(cid:12)(cid:12) e e iθ + me iθ e e iθ − (cid:12)(cid:12)(cid:12) = ( me cos θ cos( θ + sin θ ) + e θ cos(2 sin θ ) − + ( me cos θ sin( θ + sin θ ) + e θ sin(2 sin θ )) =: (cid:96) ( θ ) . The function (cid:96) attains its minimum at θ = π by applying the second derivative test andmin θ ∈ [0 , π ) g ( θ ) = g ( π ) = (cid:18) − me + 1 e − (cid:19) so that | φ ( e e iθ , e e iθ + me iθ ; z ) − | ≥ me − e + 1 ≥ e − e + 1 . Therefore φ ( e e iθ , e e iθ + me iθ ; z ) (cid:54)∈ Ω whenever z ∈ D , θ ∈ [0 , π ) and m ≥ (cid:26) w ∈ C : | w − | < e − e + 1 (cid:27) . Hence by Theorem 3.5, it follows that if f ∈ A and (cid:12)(cid:12)(cid:12)(cid:12) z ( B cκ f ( z )) (cid:48) B cκ f ( z ) (cid:18) z ( B cκ f ( z )) (cid:48)(cid:48) ( B cκ f ( z )) (cid:48) (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) < e − e + 1then B cκ f ≺ S *e . In particular, we have (cid:12)(cid:12)(cid:12)(cid:12) zϑ (cid:48) ν ( z ) ϑ ν ( z ) (cid:18) zϑ (cid:48)(cid:48) ν ( z ) ϑ (cid:48) ν ( z ) (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) < e − e + 1 = ⇒ (cid:12)(cid:12)(cid:12)(cid:12) log zϑ (cid:48) ν ( z ) ϑ ν ( z ) (cid:12)(cid:12)(cid:12)(cid:12) < z ∈ D ) . In Example 3.6, we proved that ϑ − / , , ∈ S *e . However, the functions ϑ − / , , ( z ) = z (cos √ z + √ z sin √ z )and ϑ − / , , ( z ) = z cos √ z are not members of S *e . Infact, the function ϑ − / , , does not even lie in the class S .Although the function ϑ − / , , is univalent [2, Corollary 2.9, p. 57], the figure of theimage of zϑ (cid:48)− / , , /ϑ − / , , depicted in Figure 4 clearly proves that the function ϑ − / , , does not belong to S *e . This shows that Corollary 3.2 is valid only for certain range ofvalues of ν ∈ R . |log | = 1 w - - Figure 4.
Image of zϑ (cid:48)(cid:48)− / , , ( z ) /ϑ (cid:48)− / , , ( z ) under D . References [1] A. Baricz, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen (2008),no. 1-2, 155–178.[2] A. Baricz, Generalized Bessel functions of the first kind , Lecture Notes in Mathematics, 1994,Springer-Verlag, Berlin, 2010.[3] A. Baricz, E. Deniz, M. C¸ a˘glar and H.Orhan, Differential subordinations involving generalized Besselfunctions, Bull. Malays. Math. Sci. Soc. (2015), no. 3, 1255–1280.[4] A. Baricz and S. Ponnusamy, Starlikeness and convexity of generalized Bessel functions, IntegralTransforms Spec. Funct. (2010), no. 9-10, 641–653.[5] A. Baricz and R. Sz´asz, The radius of convexity of normalized Bessel functions of the first kind,Anal. Appl. (Singap.) (2014), no. 5, 485–509.[6] N. Bohra and V. Ravichandran, On confluent hypergeometric functions and generalized Bessel func-tions, Anal. Math. (2017), no. 4, 533–545.[7] R. K. Brown, Univalence of Bessel functions, Proc. Amer. Math. Soc. (1960), 278–283.[8] J. B. Conway, Functions of one complex variable , second edition, Graduate Texts in Mathematics,11, Springer-Verlag, New York, 1978.[9] J. Dziok and H. M. Srivastava, Classes of analytic functions associated with the generalized hyper-geometric function, Appl. Math. Comput. (1999), no. 1, 1–13.[10] S. Kanas, S. R. Mondal and A. D. Mohammed, Relations between the generalized Bessel functionsand the Janowski class, Math. Inequal. Appl. (2018), no. 1, 165–178.[11] V. Madaan, A. Kumar and V. Ravichandran, Lemniscate convexity of generalized Bessel functions,Studia Sci. Math. Hungar. (2019), no. 4, 404–419. EOMETRIC PROPERTIES OF GENERALIZED BESSEL FUNCTION 17 [12] R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associ-ated with exponential function, Bull. Malays. Math. Sci. Soc. (2015), no. 1, 365–386.[13] S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications , Monographsand Textbooks in Pure and Applied Mathematics, 225, Marcel Dekker, Inc., New York, 2000.[14] S. R. Mondal and M. Al Dhuain, Inclusion of the generalized Bessel functions in the Janowski class,Int. J. Anal. , Art. ID 4740819, 8 pp.[15] A. Naz, S. Nagpal and V. Ravichandran, Star-likeness associated with the exponential function,Turkish J. Math. (2019), no. 3, 1353–1371.[16] J. K. Prajapat, Certain geometric properties of normalized Bessel functions, Appl. Math. Lett. (2011), no. 12, 2133–2139.[17] R. Sz´asz and P. A. Kup´an, About the univalence of the Bessel functions, Stud. Univ. Babe¸s-BolyaiMath. (2009), no. 1, 127–132.[18] C. M. Yan and J. L. Liu, Convolution properties for meromorphically multivalent functions involvinggeneralized Bessel functions, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. (2018), no. 1,293–299. Department of Mathematics, University of Delhi, Delhi–110 007, India
Email address : [email protected] Department of Mathematics, Ramanujan College, University of Delhi, Delhi–110 019,India
Email address : [email protected] Department of Mathematics, National Institute of Technology, Tiruchirappalli–620015, India
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