On certain multivalent functions involving the generalized Srivastava-Attiya operator

Lemma 1.5. Let q be convex in D and ℜ(µ1q(z) + µ2) > 0, where µ1, µ2 ∈C \ {0}. If h is analytic in D with q(0) = h(0) and h(z) + zh ′(z) µ1h(z) + µ2 ≺q(z), (z ∈D) , then h(z) ≺q(z). The main purpose of this paper is to derive some inclusion relationships, integral representation and arc length problems for the function classes Rs,b m,β [p, A, B, σ], Vs,b m,β [p, A, B, σ] and Ms,b m,β [p, A, B, σ, α]. The behavior of these classes under a certain integral operator is also discussed.

Theorem 2.1. Let f ∈A(p) with Js,bf(z) ̸= 0. Then Ms,b 2,β [p, A, B, σ, α] ⊂Rs,b 2,β [p, A, B, σ] .

Theorem 2.2. If f ∈A(p) with Js,bf(z) ̸= 0, z ∈D, then Rs,b 2,β [p, A, B, σ] ⊂Rs+1,b 2,β [p, A, B, σ] .

Theorem 2.3. If f ∈A(p) with Js,bf(z) ̸= 0, z ∈D, then Vs,b 2,β [p, A, B, σ] ⊂Vs+1,b 2,β [p, A, B, σ] .

Theorem 2.4. If 0 < α1 ≤α2 < 1, then Ms,b 2,β [p, A, B, σ, α2] ⊂Ms,b 2,β [p, A, B, σ, α1] .

Theorem 2.5. Let f ∈Rs,b m,β [p, A, B, σ] . If s1, s2 ∈R0,b 2,β [p, A, B, σ], then Js,bf(z) = (s1(z)) m 4 + 1 2 (s2(z)) m 4 −1 2 . (2.2)

Theorem 2.6. Let f ∈Ms,b m,β [p, A, B, σ, α] . Then g ∈Rs,b m,β [p, A, B, σ] , where Js,bg(z) z  1 p = Js,bf(z) z  1−α p  (Js,bf(z))′ α p . (2.5)

Theorem 2.7. A function f ∈Ms,b m,β [p, A, B, σ, α], if and only if there exists a function g ∈Rs,b m,β [p, A, B, σ] such that Js,bf(z) = " 1 α Z z 0 t 1 α−1 Js,bg(z) z

Theorem 2.8. Suppose that f ∈Ms,b m,0 [p, A, B, σ, α], Lr(f) denotes the length of the curve C, C = f(reiθ), 0 < θ ≤2π, and M(r) = max 0<θ≤2π f(reiθ) . Then, for 0 < r < 1, Lr(f) ≤(2 −α)πpM(r) α 2 + (k −2) A1 −kB 1 −B  , where A1 = (1 −α)A + αB.

Theorem 2.9. Let f ∈Ms,b m,0 [p, A, B, σ, α]. Then n |an| = O(1)M  1 −1 n  , (n ≥2), where O(1) is a constant depending on A1, B, p, α and k only.

Theorem 2.10. Let c be a real number with c > −p, and Js,bFc,p(z) ̸= 0, for all z ∈D. If f ∈ Rs,b 2,β [p, A, B, σ] , then Fc,p(z) ∈Rs,b 2,β [p, A, B, σ] , where Fc,p : A(p) →A(p) is defined by Fc,p(z) = c + p zc Z z 0 tc−1f(t)dt =

Theorem 2.11. Let c be a real number with c > −p, and Js,bFc,p(z) ̸= 0 for all z ∈D. If f ∈Vs,b 2,β [p, A, B, σ] , then Fc,p(z) ∈Vs,b 2,β [p, A, B, σ] , where Fc,p(z) is given by (2.8).