Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 466 https://internationalpubls.com Subclass of univalent functions involving Raducanu-Orhan differential operator connected with Pascal distribution series 1Saravanan K, 2Thirucheran M, 3Bhuvaneswari Raja, 4Stalin T, 5Britto Manoj A 1Department of Mathematics, Dr Ambedkar Government Arts College, Chennai, 600 039, India. saravanandagac@gmail.com 2Department of Mathematics, L N Government College, Ponneri, Chennai, 601 204, India. drthirucheran@gmail.com 3Department of Mathematics, Aalim Muhammed Salegh College of Engineering, Chennai, 600 055, India. rbrs1947@gmail.com 4Department of Mathematics, Vel Tech Rangarajan Dr Sagunthala R &D Institute of Science and Technology, Chennai, 600 062, India. drstalint@veltech.edu.in, https://orcid.org/0000-0002-8735-3567 5Department of Advanced Computer Science and Engineering, Vignan’s Foundation for Science, Technology & Research, Guntur- 522213, Andhra Pradesh, India. brittomanoj@gmail.com Article History: Received: 13-11-2024 Revised: 17-12-2024 Accepted: 20-01-2025 Abstract: Recent years have shown us how fascinating the univalent function is many new publications have been written in this field. Currently, operators of normalized analytic functions, differential and integral operators are highly sought after. Numerous researchers have examined and debated a great deal of material for the operators. This work introduces a new subclass 𝒫𝒬q,δ,μ n,r (θ) of the function class for univalent functions defined by the Raducanu-Orhan differential operator connected with pascal distribution series. Our goal in this work is to further our understanding and make inferences regarding the functions that are a part of these new subclass. Furthermore, the convexity of the subclass, growth and distortion, radius of starlike, extreme points, and integral means of inequalities are obtained. All this research was performed inside an open unit disc. Keywords: Analytic function, univalent function, differential operator, subordination, coefficient inequality, starlike and convexity. 1. Introduction Consider that the class 𝒜 of univalent function has the following form 𝑓(𝜉) = ξ + ∑ avξ v,∞ v=2 ξ ∈ 𝕌: = {ξ ∈ ℂ ∶ |ξ| < 1}, (1) which is analytic in the unit disc 𝕌, and 𝑔(𝜉) = ξ + ∑ bvξ v, ξϵ𝕌∞ v=2 (2) Then the convolution of (1) and (2) is represented by (𝑓 ∗ 𝑔)(𝜉) = ξ + ∑ avbvξ v,∞ v=2 ξ ∈ 𝕌 (3) Let 𝑓(𝜉) ∈ 𝐾(𝛼) then 𝑓(𝜉) is convex of order 𝛼, (0 ≤ 𝛼 < 1) in 𝕌, iff 𝑅𝑒 ( 𝜉𝑓"(𝜉) 𝑓′(𝜉 + 1) > 𝛼, 𝜉 ∈ 𝕌. Let 𝑓(𝜉) ∈ 𝑆∗(𝛼), then 𝑓(𝜉) is starlike of order 𝛼, (0 ≤ 𝛼 < 1) in 𝕌, iff mailto:saravanandagac@gmail.com mailto:drthirucheran@gmail.com mailto:rbrs1947@gmail.com mailto:drstalint@veltech.edu.in https://orcid.org/0000-0002-8735-3567 mailto:brittomanoj@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 467 https://internationalpubls.com 𝑅𝑒 ( 𝜉𝑓′(𝜉) 𝑓(𝜉) ) > 𝛼, 𝜉 ∈ 𝕌. The class 𝐾(𝛼) and 𝑆∗(𝛼) introduced by Roberston [10]. After that many authors introduced the various subclass of starlike and convex functions connected with some differential operators. The Schwarz function in 𝕌, ω(ξ) exists if and only if 𝑓(𝜉) and 𝑔(𝜉) are analytic. It is our claim that 𝑓(𝜉) is subordinate to 𝑔(𝜉); that is 𝑓(𝜉) ≺ 𝑔(𝜉). In this case, ω(0) = 0 and |ω| < 1 such that 𝑓(𝜉) = 𝑔(𝜔(𝜉)) as proven, 𝑓(𝜉) ≺ 𝑔(𝜉) and f(𝕌) ⊂ g(𝕌) Implied by 𝑓(0) = 𝑔(0). Ma and Minda [13] used the idea of subordination to create various sub classes of radii of convexity and starlikeness. To achieve this goal, a univalent function ϕ(ξ) is taken into consideration. This function is analytic and defined on 𝕌 with a positive real portion, such that ϕ′(0) > 0 and ϕ(0) =1. For 𝑓(ξ) ϵ 𝒜, Raducanu-Orhan [4] introduced the differential operator 𝒬δ,μ n 𝑓(ξ) = 𝒬δ,μ(𝒬δ,μ n−1) = ξ + ∑ [1 + (υ − 1)(δ − μ + υδμ)]n∞ υ=2 avξ υ, (4) where n ∈ ℕ0= ℕ∪0, ℕ = {1, 2, ...,}, µ, δ ≥ 0, ξ ∈ 𝕌. Remark: 𝒬δ,0 n =𝒟n yields the operator of Al-Oboudi derivative [5], 𝒬1,0 n =𝒟n is the Salagean derivative operator [7]. Recent studies have focused on a subclass of univalent functions associated with distribution series. These include the Borel, Pascal, Binomial, Poisson, Geometric, exponential, and generalized distributions as well as a generalized discrete probability distribution. In recent years, various sub class of univalent functions related to pascal distribution series have been studied by the following authors, B.A.Frasin et al.[2], S.Porwal [9], Anitha LakshmiNarayanan et al [1], G.Murugusundramoorthy [6], R.M.El-Ashwah, W.Y.Kota [10], T.Bulboaca and G.Murugusundramoorthy [12], B.A.Frasin et al. [3]. By examining the subclasses, researchers hope to gain a deeper understanding of the structure and behaviour of analytic functions, hence advancing their knowledge of complex analysis and its applications, which provides an extensive investigation of this area of study. 2. The Subclass 𝒫𝒬q,δ,μ n,r (θ) The probabilities (1 − q)r, q2r(r+1)(1−q)r 2! , qr(1−q)r 1! , q3r(r+1)(r+2)(1−q)r 3! , . .. correspond to a variable 𝓍 with values of 0,1,2, and 3, respectively, where 𝑞, and 𝑟 are called the parameters, and thus 𝒫(x = 𝓍) = ( n+r−1 r−1 ) q𝓍(1 − q)r, 𝓍 ϵ {0,1,2,3, . . . } (5) According to S.M.El-Deepa et al.[11],the power series of equation (6) is examined, with its coefficients representing probabilities of the pascal distribution, that is 𝒫q r(ξ) = ξ + ∑ ( v+r−2 r−1 ) qv−1(1 − q)rξ v, ξ ϵ 𝕌, r ≥ 1, 0 ≤ q ≤ 1∞ v=2 (6) And the linear opeerator 𝒟q r : 𝒜 → 𝒜 is defined by Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 468 https://internationalpubls.com 𝒟q r (f(ξ)) = 𝒫q r ∗ f(ξ) = ξ + ∑ ( v+2−r r−1 ) qv−1(1 − q)ravξ v , ξ ∈ 𝕌∞ v=2 (7) By using the convolution (Hadamard product) of two equations (4) and (7), the linear operator 𝒫𝒟q,δ,μ n,r f(ξ) ∶ 𝒜 → 𝒜 is defined by 𝒫𝒟q,δ,μ n,r f(ξ) = ξ + ∑ avcvξ v ,∞ v=2 (8) Where cv = [1 + (v − 1)(δ − μ + vδμ)]n ( v+r−2 r−1 ) qv−1(1 − q)r. the new subclass is defined in the following definitions: Definition 2.1. Let 𝒫𝒟q,δ,μ n,r (θ) represents a class of f in 𝒜. Then Re (1 + 1 b ( ξ(𝒫𝒟q,δ,μ n,r f(ξ))′ 𝒫𝒟 q,δ,μ n,r f(ξ) − 1)) > θ (9) Where r ≥1, 0 ≤ q ≤ 1,µ, δ ≥ 0,n ∈ ℕ0 0 ≤ θ < 1, b ∈ ℂ − {0}, and ξ ∈ 𝕌. Theorem 2.2 (Coefficient Inequalities) Let (1) define f(ξ) ∈ 𝒫𝒬q,δ,μ n,r (θ). Then ∑ ϕ v ∞ v=2 cv |av| ≤ (1 − θ)|b|, (10) Where ϕ v = |1 − b − v + θb|, cv = [1 + (v − 1)(δ − μ + vδμ)]n ( v+r−2 r−1 ) qv−1(1 − q)r, r ≥ 1,0 ≤ q ≤ 1, μ, δ ≥ 0, n ∈ ℕ0 ,0 ≤ θ < 1, b ∈ ℂ − {0}and ξ ∈ 𝕌. Proof: Let F(z) = 1 + 1 b ( ξ(𝒫𝒟q,δ,μ n,r f(ξ))′ 𝒫𝒟 q,δ,μ n,r f(ξ) − 1) – θ, = 1 + ( ξ(𝒫𝒟q,δ,μ n,r f(ξ))′ 𝒫𝒟 q,δ,μ n,r f(ξ) − 1 b ) – θ = 1 + ( ξ (𝒫𝒟q,δ,μ n,r f(ξ)) ′ − b𝒫𝒟q,δ,μ n,r f(ξ) − θb𝒫𝒟q,δ,μ n,r f(ξ) b𝒫𝒟q,δ,μ n,r f(ξ) ) By the condition of the class, F(z) ≺ 1+z 1−z . A schwarz function ω(z) exist, and ω(0) = 0 as F(z) = 1+ω(z) 1−ω(z) , where |ω| < 1. ∴ ω(z) = F(z)−1 F(z)+1 . We know that | ω(z)| = | F(z)−1 F(z)+1 | < 1. Then Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 469 https://internationalpubls.com | F(z)−1 F(z)+1 | = | ξ(𝒫𝒟q,δ,μ n,r f(ξ)) ′ −(1+θb)𝒫𝒟q,δ,μ n,r f(ξ) ξ(𝒫𝒟 q,δ,μ n,r f(ξ)) ′ −(1+θb−2b)𝒫𝒟 q,δ,μ n,r f(ξ) | = | z+∑ vcvavzv∞ v=2 −(1+θb)z−∑ (1+θb)∞ v=2 cvavzv z+∑ vcvavzv∞ k=2 −(1+θb−2b)z−∑ (1+θb−2b)cvavzv∞ v=2 | = | θb+∑ (1+θb−v)cvavzv−1∞ v=2 (2−θ)b+∑ (1+θb−2b−v)c v avzv−1∞ v=2 | ≤ | θ|b|+∑ |(1+θb−v||cvav||zv−1|∞ v=2 (2−θ)|b|−∑ |(1+θb−2b−v)||cvav||zv−1|∞ v=2 |. Which is bounded by 1, if θ|b| + ∑ |(1 + θb − v)|cv|av| ≤ (2 − θ)|b| − ∑ |(1 + θb − 2b − v)|∞ v=2 ∞ v=2 cv|av|. ∑|(1 + θb − b − v)|cv|av| ≤ (1 − θ)|b|. ∞ v=2 Hence equation (10) holds. Corollary 2.3 Let f ∈ 𝒫𝒬q,δ,μ n,r (θ) then we have av ≤ (1−θ)|b| ϕ v cv and f(ξ) = ξ + (1−θ)|b| ϕ v cv ξ v , v = 2,3,4, . .. (11) equals itself. The function f ϵ𝒜 is the subclass 𝒫𝒬q,δ,μ n,r (θ) ∁ 𝒫𝒬q,δ,μ n,r (θ), which we define. The extreme points of the subclass 𝒫𝒬q,δ,μ n,r (θ) are now determined. Theorem 2.4(Extreme points) Let f1(ξ) = ξ, fv(ξ) = ξ + ∑ η v (1−θ)|b| ϕ v cv ∞ v=2 ξ v, v ≥ 2. Then f ϵ 𝒫𝒬q,δ,μ n,r (θ) strictly if f(ξ) = ∑ η v fv(ξ)∞ v=1 , Where η v > 0 and ∑ η v = 1∞ v=1 . Proof: Let f(ξ) = ∑ η v fv(ξ) ∞ v=1 = ξ + ∑ η v (1 − θ)|b| ϕ v cv ∞ v=2 ξ v = ∑ η v (1 − θ)|b| ϕ v cv ∞ v=2 ϕ v cv Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 470 https://internationalpubls.com = (1 − θ|b| ∑ η v ∞ v=1 = (1 − θ )|b|(1 − η 1 ) < (1 − θ)|b|, Which shows that f ϵ 𝒫𝒬q,δ,μ n,r (θ) . Conversely, suppose that f ϵ 𝒫𝒬q,δ,μ n,r (θ) . since |av| ≤ (1−θ)|b| ϕ v cv , v = 2,3, .. Let η v ≤ ϕ v cv (1 − θ)|b| , η 1 = 1 − ∑ η v ∞ v=2 . Then we obtain f(ξ) = ∑ η v fv(ξ)∞ v=1 . Definition 2.5.(Little wood subordination theorem [8]) Considering that f and g in 𝕌 are analytic and that f(ξ)≺g(ξ), then ∫ |f(ξ)|μ2π 0 dθ ≤ ∫ |g(ξ)|μ dθ, μ > 0, 2π 0 and ξ = reiθ, 0 < r < 1 . Theorem 2.6(Integral means of inequalities) If f ϵ 𝒫𝒬q,δ,μ n,r (θ) and suppose that g(ξ) = ξ + (1−θ)|b|εv ϕ v cv ξ v, v = 2,3, . . . , |εv| = 1. if ω(ξ) is real it is given by (ω(ξ))v−1 = ϕ v cv (1−θ)|b|εv ∑ avξ v−1∞ v=2 . Then∫ |f(ξ)|μ2π 0 dθ ≤ ∫ |g(ξ)|μ dθ, for ξ = reiθ, 0 < r < 1, μ > 0 2π 0 . We need to demonstrate that to finish the theorem ∫ |1 + ∑ av ∞ v=2 ξ v−1| μ dθ ≤ 2π 0 ∫ |1 + (1 − θ)|b| ϵv ϕ v cv ξv−1| μ dθ. 2π 0 The Littlewood subordination theorem can be used to demonstrate that 1 + ∑ avξ v−1 ∞ v=2 ≺ 1 + (1 + θ)bϵv ϕ v cv ξ v−1. Let 1 + ∑ avξ v−1 ∞ v=2 ≺ 1 + (1 + θ)|b|ϵv ϕ v cv (ω(ξ))v−1 Therefore (ω(ξ))v−1 = ϕ v cv (1−θ)|b|ϵv ∑ avξ v−1∞ v=2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 471 https://internationalpubls.com Hence ω(0)=0. Furthermore, if fϵ𝒜 satisfy ϕ v cv ≤ (1 − θ)|b| . |ω(ξ)|v−1 = | ϕ v cv (1 − θ)|b|ϵv | ∑|av| ∞ v=2 |ξv−1| ≤ |ξ| < 1. Theorem 2.7(Convex of order 𝛉) Let f ϵ 𝒫𝒬q,δ,μ n,r (θ) . Then 𝑓 is convex of order θ in |ξ| < R3 , wℎere R3: = inf ( (1−θ)ϕ v cv v(v−θ)(1−θ)|b| ) 1 v−1, (v ≥2) (12) Proof: If |ξ| < R3 and the inequality (12) are valid, it is demonstrated that | ξf"(ξ) f′(ξ) | ≤ 1 − θ. (13) It is adequate to show that |ξ| ≤ ( (1 − θ)ϕ v cv v(v − θ)(1 − θ)|b| ) 1 v−1, (v ≥ 2). From (13), we obtain | ∑ v(v − 1)avξ v−1∞ v=2 1 + ∑ vavξ v−1∞ v=2 | ≤ 1 − θ. ∑(v(v − 1)avξ v−1 ∞ v=2 ≤ 1 + ∑ vavξ v−1 ∞ v=2 − θ − θ ∑ vav|ξ|v−1 ∞ v=2 ∑ (v2 − θv)av|ξ|v−1 ≤ (1 − θ)∞ v=2 . |ξ| ≤ ( 1−θ (v2−θv)av ) 1 v−1 , (v ≥ 2). |ξ| ≤ ( (1−θ)ϕ v cv v(v−θ)(1−θ)|b| ) 1 v−1 . Theorem 2.8 (Starlike of order 𝛉) Let f ϵ 𝒫 𝒬q,δ,μ n,r (θ) . Then f is starlike of order θ in |ξ| < R2, where R2: = inf ( (1−θ)ϕ v cv (v−θ)(1−θ)|b| ) 1 v−1 , (v ≥ 2) (14) Proof: If |ξ|