In this article, the boundedness of the generalized parametric Marcinkiewicz integral operators ${\mathcal{M}}_{\mathsf{\Omega },\varphi ,h,\rho }^{\left(r\right)}$ is considered. Under the condition that $\mathsf{\Omega }$ is a function in $L^q\left({\mathbf{S}}^{n-1}\right)$ with $q\in (1,2]$ , appropriate estimates of the aforementioned operators from Triebel–Lizorkin spaces to $L^p$ spaces are obtained. By these estimates and an extrapolation argument, we establish the boundedness of such operators when the kernel function $\mathsf{\Omega }$ belongs to the block space $B_q^{0,\nu -1}\left({\mathbf{S}}^{n-1}\right)$ or in the space $L{\left(\log L\right)}^{\nu }\left({\mathbf{S}}^{n-1}\right)$ . Our results represent improvements and extensions of some known results in generalized parametric Marcinkiewicz integrals.

For normalized analytic functions f(z) with f(z) ≠ 0 for 0 < |z| < 1, we introduce a univalence criterion defined by sharp inequality associated with the nth derivative of z/f(z), where n ∈ {3,4, 5, …}.

Let q ₁, q ₂ be univalent in Δ:= {z: |z| < 1} and p be certain analytic function. We give some applications of first order differential subordinations and superordinations to obtain sufficient conditions to satisfy the following sandwich implication which is a generalization for various known sandwich theorems: $$ \beta zq_1^k (z)q'_1 (z) + \sum\limits_{j = 0}^n {\alpha _j q_1^j (z)} \prec \beta zp^k (z)p'(z) + \sum\limits_{j = 0}^n {\alpha _j p^j (z)} \prec \beta zq_2^k (z)q'_2 (z) + \sum\limits_{j = 0}^n {\alpha _j q_2^j (z)} $$ implies q ₁(z) ≺ p(z) ≺ q ₂(z), where k ∈ ℤ and β ≠ 0, α′ _j ∈ ℂ. Some of its special cases and its applications will be considered for certain analytic functions and certain linear operators.

In this article, certain sharp $L^p$ estimates for a specific class of generalized Marcinkiewicz operators with mixed homogeneity associated to surfaces of revolution are established. By virtue of Yano’s extrapolation argument, beside these estimates, the $L^p$ boundedness of the aforementioned operators under weaker assumptions on the kernels is confirmed. The obtained results in this article are fundamental extensions and improvements of numerous previously known results on parabolic generalized Marcinkiewicz integrals.