In this paper, by the fixed point method, we prove the generalized stability of random homomorphisms of the following additive-quadratic functional equation f(2 + y) + f(2 − y) = 2[f(x + y) + f(x − y)] + 2[f(x) + f(−x)] − [f(y) + f(−y)], in random normed algebras.

The articles introduces and investigates two new subclasses of the bi-univalent functions These are analytical functions related to the m-fold symmetric function We calculate the initial coefficients for all the functions that belong to them, as well as the coefficients for the functions that belong to a field where finding these coefficients requires a complicated method. Between the remaining results, the upper bounds for the initial coefficients are found in our study as well as several examples. We also provide a general formula for the function and its inverse in the m-field. A function is called analytical if it does not take the same values twice It is called a univalent function if it is analytical at all its points, and the function is called a bi-univalent if it and its inverse are univalent functions together. We also discuss other concepts and important terms.

In this paper, we introduce and investigate two subclasses M_Σ^q (λ,γ,h) and H_Σ^q (η,δ,h) of bi-univalent functions defined by quasi-subordination. We find estimates on the Taylor-Maclaurin coefficients |a_2 | and |a_3 | for functions in these subclasses.

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This study presents the notion of neutrosophic Z-algebra and neutrosophic pseudo Z-algebra explores some of its properties. Also studied are the neutrosophic Z-ideal, neutrosophic Z-sub algebra, and neutrosophic Z-filter. Several properties are discovered, and some findings from the study of homomorphism are discussed.