Abstract

In this paper, by means of a q−difference operator, we first introduce a q−p− Opoola differential operator.

(𝑝, 𝑟, 𝜆 ∈ 𝑁 ∶= {1, 2,· · · }; 𝑛,𝑡 ∈ 𝑁 ∪ 0 ∶= {0, 1, 2,· · · }), which is related to 𝐷 𝑛(𝛾,𝜑,𝑡)𝑓(𝑧) = 𝑧 + ∑ [1 + ∞ 𝑘=2 (𝑘 +𝜑 − 𝛾 − 1)𝑡] 𝑛𝑎𝑘𝑧 𝑘 , 𝑧 ∈ 𝑈 0 ≤ 𝛾 ≤ 𝜑, 𝑡 ≥ 0 when 𝑝 = 𝑟 = 1 in 𝐹(𝑧) ∈ 𝑇(𝑝, 𝑟). Via employing the q−p− Opoola differential operator, we define a new family of close-to-convex functions and obtain a coefficient estimate theorem. Furthermore, we also introduced several modified-Hadamard products for the family and finally gave some corollaries. 

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