arXiv:2007.05439v1 [math.CV] 10 Jul 2020 eeaigplnmas(e 3,[5,[7)o elpolyno Bell or [17]) [15], [3], (see polynomials generating [5 functions hypergeometric Gaussian generalized, variou the of for properties geometric and conject analytical Bieberbach with famous dealing the of solution the in [6] generali branc Branges of various use in surprising The applied Technology. frequently and Engineering is It research. of topic h ocadplnmas tde yJcusTuhr [19] Touchard Jacques by studied polynomials, Touchard The T Function Geometric on function special of application The ocadplnmas series. polynomials. Touchard ewrsadPhrases and Keywords 30C45. Classification: Subject Mathematics 2000 nltcuiaetfntoswt oiiececet nt in coefficients positive with su investigate functions univalent we analytic precise, more cer be applying To by polynomials. Touchard functions univalent analytic of subclasses Abstract. NVLN UCIN IHPSTV COEFFICIENTS POSITIVE WITH FUNCTIONS UNIVALENT G.MURUGUSUNDARAMOORTHY h ups ftepeetppri oetbihconnections establish to is paper present the of purpose The NOVN OCADPOLYNOMIALS TOUCHARD INVOLVING ar-hvaprKnu-025 UP) India (U.P.), Bairi-Shivrajpur-Kanpur-209205, nvln,Salk ucin,Cne ucin,Hadama functions, Convex functions, Starlike Univalent, : a aa oenetDge College, Degree Government Sahai Ram 2 1 email:[email protected] , colo dacdSciences, Advanced of School ∗ eateto Mathematics of Department 1. elr 304 India. 632014, - Vellore INTRODUCTION I University, VIT 1 1 N ARB PORWAL SAURABH AND eoe ntdisk unit open he ancnouinoeao involving operator convolution tain 0 1 6 18]. 16, 11, 10, , il se[] opieapolynomial a comprise [1]) (see mials ye fseilfntos especially functions, special of types s e yegoercfnto yL de L. by function hypergeometric zed e ahmtc,Pyis Sciences, Physics, Mathematics, hes hcnetoswt h lse of classes the with connections ch r.Teei netnieliterature extensive an is There ure. er sacretadinteresting and current a is heory locle h exponential the called also , U . ewe various between 2 , ∗ dproduct, rd UNIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS... 2 sequence of that for X is a random variable with a with expected value ℓ, then its nth moment is E(Xκ)= T(κ, ℓ), leading to the form: ∞ κnnℓ T(κ, ℓ)= eκ zn (1.1) n! Xn=0 Touchard polynomials coefficients after the second force as following

Lately, introduce Touchard polynomials coefficients after the second force as following ∞ ℓ n−1 (n − 1) κ − Φℓ (z)= z + e κzn, z ∈ D, (1.2) κ (n − 1)! nX=2 where ℓ ≥ 0; κ > 0 and we note that, by ratio test the radius of convergence of above series is infinity. Let H be the class of functions analytic in the unit disk U = {z ∈ C : |z| < 1}. Let A be the class of functions f ∈ H of the form ∞ n f(z)= z + anz , z ∈ U. (1.3) nX=2 We also let S be the subclass of A consisting of functions which are normalized by f(0) = 0 = f ′(0) − 1 and also univalent in U.

Denote by V the subclass of A consisting of functions of the form ∞ n f(z)= z + anz , an ≥ 0. (1.4) nX=2 A A ∞ n For functions f ∈ given by (1.3) and g ∈ given by g(z) = z + n=2 bnz , we define the Hadamard product (or convolution) of f and g by P ∞ n (f ∗ g)(z)= z + anbnz , z ∈ U. (1.5) nX=2

Now, we define the linear operator

I(l,m,z) : A → A defined by the convolution or hadamard product ∞ l n−1 ℓ (n − 1) m −m n I(l,m,z)f =Φ (z) ∗ f(z)= z + e anz , (1.6) m (n − 1)! nX=2 ℓ where Φm(z) is the series given by (1.2). M 4 The class (α) of starlike functions of order 1 < α ≤ 3 zf ′(z) M(α) := f ∈ A : ℜ < α, z ∈ U  f(z)  UNIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS... 3

N 4 and the class (α) of convex functions of order 1 < α ≤ 3 ′′ zf (z) ′ N(α):= f ∈ A : ℜ 1+ < α, z ∈ U = f ∈ A : zf ∈ M(α)   f ′(z)    were introduced by Uralegaddi et al.[20] (see[7, 9]).Also let M∗(α) ≡ M(α) ∩ V and N∗(α) ≡ N(α) ∩ V.

In this paper we introduce two new subclasses of S namely M(λ, α) and N(λ, α) to discuss some inclusion properties. 4 M N For some α (1 < α ≤ 3 ) and λ(0 ≤ λ< 1), we let (λ, α) and (λ, α) be two new subclass of S consisting of functions of the form (1.3) satisfying the analytic criteria

zf ′(z) M(λ, α) := f ∈ S : ℜ < α, z ∈ U . (1.7)  (1 − λ)f(z)+ λzf ′(z) 

f ′(z)+ zf ′′(z) N(λ, α) := f ∈ S : ℜ < α, z ∈ U . (1.8)  f ′(z)+ λzf ′′(z)  We also let M∗(λ, α) ≡ M(λ, α) ∩ V and N∗(λ, α) ≡ N(λ, α) ∩ V.

Note that M(0, α) = M(α), N(0, α) = N(α); M∗(α) and N∗(α) the subclasses of studied by Uralegaddi et al.[20].

Motivated by results on connections between various subclasses of analytic univalent functions by using hypergeometric functions (see [5, 10, 11, 16, 18]), we obtain necessary and sufficient condition for function Φ(l,m,z) to be in the classes M(λ, α) , N(λ, α) and connections between Rτ (A, B) by applying convolution operator.

2. Preliminary Results

To prove our main results we shall require the following definitions and lemmas.

Definition 2.1. The lth moment of the Poisson distribution is defined as

∞ l n ′ n m − µ = e m. l n! nX=0

Lemma 2.1. 4 V M∗ For some α (1 < α ≤ 3 ) and λ(0 ≤ λ < 1), and if f ∈ then f ∈ (λ, α) if and only if ∞

[n − (1 + nλ − λ)α]|an|≤ α − 1. (2.9) nX=2 UNIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS... 4

Lemma 2.2. 4 V N∗ For some α (1 < α ≤ 3 ) and λ(0 ≤ λ< 1), and if f ∈ then f ∈ (λ, α) if and only if ∞

n[n − (1 + nλ − λ)α]an ≤ α − 1. (2.10) nX=2

3. Main Results

For convenience throughout in the sequel, we use the following notations: ∞ mn−1 = em − 1 (3.11) (n − 1)! nX=2 ∞ mn−1 = mem (3.12) (n − 2)! nX=2 ∞ mn−1 = m2em (3.13) (n − 3)! nX=2 (3.14)

∗ Theorem 3.1. If m> 0(m 6= 0, −1, −2,... ), l ∈ N0 then Φ(l,m,z) ∈ M (λ, α) if and only if

′ ′ − (1 − αλ) µ + (1 − α) µ , l ≥ 1 e m  l+1 l ≤ α − 1. (3.15)  (1 − αλ) mem + (1 − α) (em − 1) , l = 0  Proof. To prove that Φ(l,m,z) ∈ M∗(λ, α), then by virtue of Lemma 2.1, it suffices to show that

∞ l n−1 (n − 1) m − [n − (1 + nλ − λ)α] e m ≤ α − 1. (3.16) (n − 1)! nX=2 Now ∞ l n−1 − (n − 1) m e m [n(1 − λα) − α(1 − λ)] (n − 1)! Xn=2 ∞ l n−1 − (n − 1) m = e m [(n − 1)(1 − λα) + 1 − α] (n − 1)! Xn=2 ∞ l n − n m = e m [n(1 − λα) + 1 − α] n! Xn=1 ∞ l+1 n l n − n m n m = e m [(1 − λα) + (1 − α) ] n! n! Xn=1 ′ ′ − (1 − αλ) µ + (1 − α) µ , l ≥ 1 = e m  l+1 l  (1 − αλ) mem + (1 − α) (em − 1) , l = 0  UNIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS... 5

But this expression is bounded above by α − 1 if and only if (3.15) holds. Thus the proof is complete. 

∗ Theorem 3.2. If m> 0(m 6= 0, −1, −2,... ), l ∈ N0 then Φ(l,m,z) ∈ N (λ, α) if and only if

′ ′ ′ − (1 − αλ) µ + (2 − αλ − α) µ + (1 − α) µ , l ≥ 1 e m  l+2 l+1 l ≤ α − 1. (3.17)  (1 − αλ) (m2 + m)em + (2 − αλ − α) mem + (1 − α) (em − 1) , l = 0  Proof. To prove that Φ(l,m,z) ∈ N∗(λ, α), then by virtue of Lemma 2.2, it suffices to show that

∞ l n−1 (n − 1) m − n[n − (1 + nλ − λ)α] e m ≤ α − 1. (3.18) (n − 1)! nX=2 Now ∞ l n−1 − (n − 1) m e m n[n(1 − λα) − α(1 − λ)] (n − 1)! nX=2 ∞ l n−1 − (n − 1) m = e m [(1 − λα)(n − 1)2 + (2 − αλ − α)(n − 1) + 1 − α] (n − 1)! nX=2 ∞ l+2 n−1 l+1 n−1 l n−1 − (n − 1) m (n − 1) m (n − 1) m = e m [(1 − λα) + (2 − αλ − α) + (1 − α) ] (n − 1)! (n − 1)! (n − 1)! nX=2 ′ ′ ′ − (1 − αλ) µ + (2 − αλ − α) µ + (1 − α) µ , l ≥ 1 = e m  l+2 l+1 l  (1 − αλ) (m2 + m)em + (2 − αλ − α) mem + (1 − α) (em − 1) , l = 0  But this expression is bounded above by α − 1 if and only if (3.17) holds. Thus the proof is complete. 

4. Inclusion Properties

A function f ∈ A is said to be in the class Rτ (A, B), (τ ∈ C\{0}, −1 ≤ B < A ≤ 1), if it satisfies the inequality f ′(z) − 1 ′ < 1 (z ∈ U). (A − B)τ − B[f (z) − 1]

Rτ The class (A, B) was introduced earlier by Dixit and Pal [8]. It is of interest to note that if

τ = 1, A = β and B = −β (0 < β ≤ 1), we obtain the class of functions f ∈ A satisfying the inequality f ′(z) − 1 < β (z ∈ U) f ′(z) + 1

UNIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS... 6 which was studied by (among others) Padmanabhan [13] and Caplinger and Causey [4].

Lemma 4.3. [8] If f ∈ Rτ (A, B) is of form (1.3), then |τ| |a |≤ (A − B) , n ∈ N \ {1}. (4.1) n n The result is sharp.

Making use of the Lemma 4.3 we will study the action of the Poissons distribution series on the class M(λ, α).

τ Theorem 4.3. Let m > 0(m 6= 0, −1, −2,... ), l ∈ N0. If f ∈ R (A, B),then I(l,m,z)f ∈ N∗(λ, α) if and only if

′ ′ − (1 − αλ) µ + (1 − α) µ , l ≥ 1 (A − B)|τ|e m  l+1 l ≤ α − 1. (4.2)  (1 − αλ) mem + (1 − α) (em − 1) , l = 0  Proof. Let f be of the form (1.3) belong to the class Rτ (A, B). By virtue of Lemma 2.2, it suffices to show that ∞ (n − 1)lmn−1 n[n − (1 + nλ − λ)α] |an|≤ α − 1. (n − 1)! nX=2 Let Now ∞ l n−1 −m (n − 1) m e [n(1 − λα) − α(1 − λ)] |an| (n − 1)! Xn=2 ∞ l n−1 − (n − 1) m ≤ (A − B)|τ| e m [(n − 1)(1 − λα) + 1 − α] (n − 1)! Xn=2 ∞ l n − n m = (A − B)|τ|e m [n(1 − λα) + 1 − α] n! nX=1 ∞ l+1 n l n − n m n m = (A − B)|τ|e m [(1 − λα) + (1 − α) ] n! n! nX=1 ′ ′ − (1 − αλ) µ + (1 − α) µ , l ≥ 1 = (A − B)|τ|e m  l+1 l  (1 − αλ) mem + (1 − α) (em − 1) , l = 0 ≤ α − 1. 



Theorem 4.4. L z I(l,m,t) Let m > 0(m 6= 0, −1, −2,... ), l ∈ N0 then (l,m,z) = 0 t dt is in N∗(λ, α) if and only if inequality (3.15) is satisfied. R UNIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS... 7

Proof. Since ∞ l n−1 n (n − 1) m − z L(l,m,z)= z + e m . (n − 1)! n nX=2 By virtue of Lemma 3.1, it suffices to show that ∞ l n−1 (n − 1) m − n[n − (1 + nλ − λ)α] e m ≤ α − 1. n(n − 1)! Xn=2 Now, ∞ ∞ l n−1 l n−1 (n − 1) m − (n − 1) m − n[n − (1 + nλ − λ)α] e m = [n − (1 + nλ − λ)α] e m. n(n − 1)! (n − 1)! nX=2 Xn=2

Proceeding as in Theorem 3.1 we obtain the required result. 

5. Open Problems

It is interesting to find the result of Theorems 3.1-4.4, when l is a real number.

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