In this article, the boundedness of the generalized parametric Marcinkiewicz integral operators M Ω , ϕ , h , ρ ( r ) is considered. Under the condition that Ω is a function in L q ( S n - 1 ) with q ∈ ( 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 Ω belongs to the block space B q 0 , ν - 1 ( S n - 1 ) or in the space L ( l o g L ) ν ( S n - 1 ) . Our results represent improvements and extensions of some known results in generalized parametric Marcinkiewicz integrals.

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Let Bp(α, β, λ; j) be the class consisting of functions f (z) = z p + ∑ ∞ k=p+1 a k z k , p ∈ N which satisfy Re { α f (j) (z) z p−j + β f (j+1) (z) z p−j−1 + (β − α 2) f (j+2) (z) z p−j−2 } > λ, (z ∈ U = {z : |z| < 1}), for some λ (λ < p!{α+(p−j)β +(p−j)(p−j −1)(β −α)/2}/(p−j)!) and j = 0, 1, ..., p , where p+1−j +2α/(β −α) > 0 or α = β = 1. The extreme points of Bp(α, β, λ; j) are determined and various sharp inequalities related to Bp(α, β, λ; j) are obtained. These include univalence criteria, coefficient bounds, growth and distortion estimates and bounds for certain linear operators. Furthermore, inclusion properties are investigated and estimates on λ are found so that functions of Bp(α, β, λ; j) are p-valent starlike in U. For instance, Re{zf ′′ (z)} > (5 − 12 ln 2)/(44 − 48 ln 2) ≈ −0.309 is sufficient condition for any normalized analytic function f to be starlike in U. The results improve and include a number of known results as their special cases.

For normalized analytic functionsf(z)withf(z)≠0for0<|z|<1, we introduce a univalence criterion defined by sharp inequality associated with thenth derivative ofz/f(z), wheren∈{3,4,5,…}.

Problem statement:We introduced a new bijective convolution linear operator defined on the class of normalized analytic functions. This operator was motivated by many researchers namely Srivastava, Owa, Ruscheweyh and many others. The operator was essential to obtain new classes of analytic functions. Approach: Simple technique of Ruscheweyh was used in our preliminary approach to create new bijective convolution linear operator. The preliminary concept of Hadamard products was mentioned and the concept of subordination was given to give sharp proofs for certain sufficient conditions of the linear operator aforementioned. In fact, the subordinating factor sequence was used to derive different types of subordination results. Results: Having the linear operator, subordination theorems were established by using standard concept of subordination. The results reduced to wellknown results studied by various researchers. Coefficient bounds and inclusion properties, growth and closure theorems for some subclasses were also obtained. Conclusion: Therefore, many interesting results could be obtained and some applications could be gathered.