Research Article Inequalities Involving Essential Norm Estimates of Product-Type Operators Manisha Devi , Ajay K. Sharma, and Kuldip Raj School of Mathematics, Shri Mata Vaishno Devi University, Katra 182320, Jammu and Kashmir, India Correspondence should be addressed to Kuldip Raj; kuldipraj68@gmail.com Received 11 September 2020; Revised 22 January 2021; Accepted 15 February 2021; Published 3 March 2021 Academic Editor: Ming-Sheng Liu Copyright © 2021 Manisha Devi et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Consider an open unit disk D � z ∈ C: |z| < 1 { } in the complex plane C, ξ a holomorphic function on D,and ψ a holomorphic self- map of D. For an analytic function f, the weighted composition operator is denoted and defined as follows: (W ξ,ψ f)(z)� ξ(z)f(ψ(z)). We estimate the essential norm of this operator from Dirichlet-type spaces to Bers-type spaces and Bloch-type spaces. 1. Introduction and Preliminaries Consider an open unit disk D � z ∈ C: |z| < 1 { } in the complex plane C. Let H(D) denote the class of all analytic functions on D, S(D) be the class of all holomorphic self- maps of D,and H ∞ be the space of all bounded holomorphic functions on D. Let ξ ∈ H(D) and ψ be a holomorphic self- map of D. For z ∈ D, the composition operator and mul- tiplication operator are, respectively, defined by C ψ f  (z)� f(ψ(z)), M ξ f  (z)� ξ(z)f(z), f ∈ H(D). (1) e weighted composition operator is denoted and defined as W ξ,ψ f  (z)� ξ(z)f(ψ(z)), f ∈ H(D), (2) where W ξ,ψ is a product-type operator as W ξ,ψ � M ξ C ψ . Clearly, this operator can be seen as a generalization of the composition operator and multiplication operator. It can be easily seen that, for ξ ≡ 1, the operator reduced to C ψ . If ψ(z)� z, the operator gets reduced to M ξ . is operator is basically a linear transformation of H(D) defined by (W ξ,ψ f)(z)� ξ(z)f(ψ(z)) � (M ξ C ψ f)(z), for f in H(D) and z in D. e basic aim is to give the operator-theoretic characterization of these operators in terms of function- theoretic characterization of their including functions. Various holomorphic function spaces on various domains have been studied for the boundedness and compactness of weighted composition operators acting on them. Moreover, a number of papers have been studied on these operators acting on different spaces of holomorphic functions on various domains. For more details, see [1–14] and the ref- erences therein. We say that a linear operator is bounded if the image of a bounded set is a bounded set. Moreover, a linear operator is said to be compact if it maps the bounded sets to those sets whose closure is compact. For each α > 0, the weighted Bloch space B is defined as follows: B � f ∈ H(D): sup z∈D 1 −|z| 2   α f ′ (z)         <∞  . (3) In this expression, seminormed is defined. is space forms a Banach space with the natural norm defined by ‖f‖ B �|f(0)| + sup z∈D 1 −|z| 2   α f ′ (z)        . (4) For α � 1, this space gets reduced to classical Bloch space. A function ω: D ⟶(0, ∞) is said to be a weight if it is continuous. For z ∈ D, the weight ω is said to be radial if ω(z)� ω(|z|). A weight ω is said to be a standard weight if Hindawi Journal of Mathematics Volume 2021, Article ID 8811309, 9 pages https://doi.org/10.1155/2021/8811309