Starlike Functions Related to the Bell Numbers

The present paper aims to establish the first order differential subordination relations between functions with a positive real part and starlike functions related to the Bell numbers. In addition, several sharp radii estimates for functions in the class of starlike functions associated with the Bell numbers are determined.

It should be noted that the special cases of ϕ, mentioned above, are univalent in the unit disk.In 2011, Dziok et al. [13,14] considered ϕ to be a non-univalent function associated with the Fibonacci numbers, defined by which maps the unit disk D on to a shell-like domain in the right-half plane.Further, they defined the class S * The functions f ∈ S * F are starlike of order √ 5/10.Motivated by the above defined classes, we consider a function associated with the Bell Numbers.For a fixed non-negative integer n, the Bell numbers B n count the possible disjoint partitions of a set with n elements into non-empty subsets or, equivalently, the number of equivalence relations on it.The Bell numbers B n satisfy a recurrence relation involving binomial coefficients and B 6 = 203.For more details, see [15][16][17][18][19][20][21].Kumar et al. [22] considered the function which is starlike with respect to 1 and it's coefficients generate the Bell numbers.Kumar et al. [22] defined the class S * B by S * B := S * (Q).From [1], note that the function f ∈ S * B if and only if there exists an analytic function q, satisfying q(z) ≺ Q(z) (z ∈ D), such that The above representation shows that the functions in the class S * B can be seen as an integral transform I(q(z)) of the function q with f (0) = 0 and f (0) = 1.The reader may refer to the paper [23] and the references cited therein for integral transform related works.The authors in [22] determined sharp coefficient bounds on the six initial coefficients, Hankel determinant, and on the first three consecutive higher order Schwarzian derivatives for functions in the class S * B .Let P be the class of analytic functions p : D → C with p(0) = 1 and Re p(z) > 0 (z ∈ D).In 1989, Nunokawa et al. [24] showed that if 1 + zp (z) ≺ 1 + z, then p(z) ≺ 1 + z.In 2007, Ali et al. [25] computed the condition on β, in each case, for which . Further, Ali et al. [26] determined some sufficient conditions for normalized analytic functions to lemniscate starlike functions.Recently, Kumar and Ravichandran [27] obtained sufficient conditions for first order differential subordinations so that the corresponding analytic function belongs to the class P. In 2016, Tuneski [28] gave a criteria for analytic functions to be Janowski starlike.For more details, see [11,[29][30][31][32][33].Motivated by above works, in Section 2, using the theory of differential subordination developed by Miller and Mocanu, a sharp bound on parameter β is determined in each case so that p(z) ≺ Q(z), whenever 1 + βzp (z)/p j (z)(j = 0, 1, 2) is subordinate to the function ϕ 0 (z) or √ 1 + z or G α (z) or (1 + Az)/(1 + Bz) or ϕ s (z) or ϕ q (z).Further, various sufficient conditions are obtained for f ∈ A to be in the class S * B as an application of these subordination results.In Section 3, S * B -radius for the class of Janowski starlike functions and some other well-known classes of analytic functions are investigated.
Suppose that If p is analytic in D, with p(0) = q(0), p(D) ⊆ U and then p ≺ q, and q is most dominant.
Then, the following are sufficient for p(z) ≺ Q(z). (a The lower bound on β in each case is sharp.
Proof.Let the functions ν and ψ be defined by ν(w) = 1 and ψ(w It can be easily seen that Q is starlike in D and the function h is defined by Therefore, from Lemma 1, we conclude that Now the subordination p ≺ Q holds if subordination q β ≺ Q.Thus, the subordination hold and these yield a necessary condition for subordination p ≺ Q to hold.In view of the graph of the respective function, the necessary condition is also sufficient condition.The inequalities where Therefore, in view of the subordination relation 1, the required subordination p ≺ Q holds if subordination q β ≺ Q holds.Thus, the subordination q β ≺ Q holds if the inequalities hold which in-turn yield a necessary condition for subordination p ≺ Q.The inequalities q β (−1) ≥ Q(−1) and q β (1) )/(e (1−e)/e − 1), respectively.Therefore, the subordination (c) The analytic function (d) Consider the analytic function which is a solution of differential equation where .
(e) The differential equation dq dz = sin z βz has an analytic solution where ≈ 2.01905 and has an analytic solution Computation shows that the function where ≈ 1.65198 and Note that the function Q(z) = zq β (z)ψ(q β (z)) = e z is starlike in the unit disk D and the function h(z) where This ends the proof.
Theorem 1 also provides the following various sufficient conditions for the normalized analytic functions f to be in the class S * B .Let function f ∈ A and set If either of the following subordination holds The next result gives sharp lower bound on β such that subordination p ≺ Q holds, whenever 1 Theorem 2. Let 0 < α < 1, 0 < B < A < 1, and p be an analytic function defined in D with p(0) = 1. Set Then, the following conditions are sufficient for subordination p ≺ Q.
The lower bound on β in each case is sharp.
(a) The function Now using Theorem 1 (a), the subordination where and (b) The function Moreover, the function )/(e − 1).(c) Consider the function q β defined by It can be verified that the function q β is a solution of the differential equation where (d) Let the function q β (z) = exp ((A − B) log(1 + Bz)/βB) be an analytic solution of the differential equation .
(e) The differential equation βzq (z)/q(z) = sin z has an analytic solution given by .
(f) The solution of the differential equation is given by As in proof of Theorem 2 (a), the desired result holds if β ≥ max{β 1 , + log 2)/(e − 1).(g) The differential equation βzq (z)/q(z) = e z − 1 has a solution where This ends the proof.
Next, Theorem 2 also provides the following various sufficient conditions for the normalized analytic functions f to be in the class S * B .Let the function f ∈ A and set If either of the following subordination conditions are fulfilled: ≺ e z (β ≥ 0.766987), then f ∈ S * B .In the following theorem, the sharp lower bound on β is obtained so that the subordination p ≺ Q holds, whenever 1 or ϕ s (z) or ϕ q (z) or e z .These results can be proved by defining the functions ν, ψ : D → defined by ν(w) = 1 and ψ(w) = β/w 2 and proceeding in a similar fashion as in the proofs of Theorems 1 and 2. Theorem 3. Let 0 < α < 1, 0 < B < A < 1, and p be an analytic function defined in D with p(0) = 1. Set Then, the following conditions are sufficient for p ≺ Q.
The lower bound on β in each case is sharp.
Let f ∈ A and set If either of the following subordination holds ), ≺ e z (β ≥ 1.60597), then f ∈ S * B .
The main technique involved in tackling the S * B -radius estimates for classes of functions f is the determination of the disk that contains the values of z f (z)/ f (z).The associated technical lemma is achieved as: Then, the following holds: Proof.To prove the assertion, we let z = e it , t ∈ (−π, π].Therefore, Q(e it ) = e e e it −1 = u(t) + iv(t) with u(t) := cos sin(sin t)e cos t exp e cos t cos(sin t) − 1 and v(t) := sin sin(sin t)e cos t exp e cos(t) cos(sin t) − 1 .
Now, consider the square of the distance of an arbitrary point (u(t), v(t)) on the boundary of ∂Q(D) from (a, 0) and is given by h(t) = d 2 (t) = a 2 − 2ae e cos t cos(sin t)−1 cos sin(sin t)e cos t + e 2e cos t cos(sin t)−2 .Now we need to prove |w − a| < r(a) is the largest disk contained in Q(D).For this, we need to show that min −π≤t≤π d(t) = r(a).Since h is an even function, i.e., h(t) = h(−t), we need to only consider the case when t ∈ [0, π].Now h (t) = 0 has three roots viz.0, π and t 0 (a) ∈ (0, π).Among these roots, the root t 0 (a) depends on a and graphics reveals that h is increasing in the interval [0, t 0 (a)] and decreasing in [t 0 (a), π], and therefore, h attains its minimum either at 0 or π.Further computations give h(π) = ea − e 1/e 2 /e 2 and h(0) = (e e − ea) 2 /e 2 .Hence, we have ≤ a ≤ e e−1 .
To find the circle of minimum radius with center at (1, 0) containing the domain Q(D), we need to find the maximum distance from (1, 0) to an arbitrary point on the boundary of the domain Q(D).The square of this distance function is given by φ(t) = −2e e cos t cos(sin t)−1 cos sin(sin t)e cos t + e 2e cos t cos(sin t)−2 + 1.
Hence, the radius of the smallest disk containing Q(D) is (e − e e ) /e.This ends the proof.
We now recall some classes and results related to them which are to be used for further development of this section.For −1 ≤ respectively.These classes were introduced and studied by [2].Further, let S * (α The following results will be needed: In particular, if p ∈ P n (α), then, for |z| = r, The main objective of this section is to determine the S * B -radii constants for functions belonging to certain well-known subclasses of A. Let G denote the class of functions f ∈ S for which f (z)/z ∈ P.
The following theorem gives the sharp S * B -radius for the class G.Because the function k is univalent too, it follows that the S * B -radius for the class S and S * is r 4 .Therefore, the radius r 4 can not be increased.Thus, we have the following:  Then, through some assumptions, we have p, h ∈ P.

Corollary 1 .
The sharp S * B -radius for the classes S and S * is (e − e 1/e )/(e + e 1/e ) ≈ 0.30594.Let the class F 1 be defined by