Tom 28 (2020): Zeszyt 1 (June 2020)

In this paper, we find conditions under which the bracket defined by a graded derivation on a Lie superalgebra (g, [, ]) is skew-supersymmetry and satisfies the super Jacobi identity, so it defines the structure of a Lie superalgebra on g.

In the case of the algebra of differential forms on a supermanifold, we study the graded commutator of graded derivations, graded skew-derivations and a graded derivation, with another graded skew-derivation of the superalgebra of differential forms on a supermanifold.

In the year 2006, S. Lin and W. Lin introduced the definition of weakly weighted-sharing of meromorphic functions which is between “CM” and “IM”. In this paper, using the notion of weakly weighted-sharing, we study the uniqueness of a polynomial function p(f) of f and a homogeneous differential polynomial P [f] generated by f. Our results improve and generalizes the results due to Charak and Lal, S. Lin and W. Lin, and H-Y Xu and Y Hu.

In this study, the authors establish and generalize some inequalities of Hadamard and Simpson type based on s-convexity in the second sense. Some applications are also given and generalized. Examples are given to show the results. The results generalize the integral inequalities in articles of Sarikaya and Xi.

In this paper, we give some common fixed point theorems for a class of occasionally weakly compatible mappings satisfying contractive conditions of integral type. Our results generalize a host of previously theorems. We also present some illustrative examples which support our main results and show the applicability and validity of these results.

If f, g: [a, b] → 𝕉 are two continuous functions, then there exists a point c ∈ (a, b) such that

∫acf(x)dx+(c-a)g(c)=∫cbg(x)dx+(b-c)f(c).\int_a^c {f\left(x \right)} dx + \left({c - a} \right)g\left(c \right) = \int_c^b {g\left(x \right)} dx + \left({b - c} \right)f\left(c \right).

In this paper, we study the approaching of the point c towards a, when b approaches a.

Assume that f and g are continuous on γ, γ ⊂ 𝔺 is a piecewise smooth path parametrized by z (t), t ∈ [a, b] from z (a) = u to z (b) = w with w ≠ u and the complex Čebyšev functional is defined by

𝒟γ(f,g):=1w-u∫γf(z)g(z)dz-1w-u∫γf(z)dz1w-u∫γg(z)dz.{{\cal D}_\gamma}\left({f,g} \right): = {1 \over {w - u}}\int_\gamma {f\left(z \right)} g\left(z \right)dz - {1 \over {w - u}}\int_\gamma {f\left(z \right)} dz{1 \over {w - u}}\int_\gamma {g\left(z \right)} dz.

In this paper we establish some Grüss type inequalities for 𝒟 (f, g) under some complex boundedness conditions for the functions f and g.

In this paper, we study a class of univalent functions f as defined by making use of the generalized Ruscheweyh derivatives involving a general fractional derivative operator, satisfying

Re{z(J1λ,μf(z))'(1-γ)J1λ,μf(z)+γz2(J1λ,μf(z))''}>β.{\mathop{\rm Re}\nolimits} \left\{{{{z\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)'} \over {\left({1 - \gamma} \right){\bf{J}}_1^{\lambda,\mu}f\left(z \right) + \gamma {z^2}\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)''}}} \right\} > \beta.

A necessary and sufficient condition for a function to be in the class Aγλ,μ,ν(n,β)A_\gamma ^{\lambda,\mu,\nu}\left({n,\beta} \right) is obtained. Also, our paper includes linear combination, integral operators and we introduce the subclass Aγ,cmλ,μ,ν(1,β)A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right) consisting of functions with negative and fixed finitely many coefficients. We study some interesting properties of Aγ,cmλ,μ,ν(1,β)A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right).

Inequality study is a magnificent field for investigating the geometric behaviors of analytic functions in the open unit disk calling the subordination and superordination. In this work, we aim to formulate a generalized differential-difference operator. We introduce a new class of analytic functions having the generalized operator. Some subordination results are included in the sequel.

In the present paper, we find certain results on Ruscheweyh operator using differential inequality. In particular, we find sufficient conditions for starlike and convex functions.

This paper is concerned with certain subclasses of univalent and bi-univalent functions related to shell-like curves connected with Fibonacci numbers. We find estimates of the initial coefficients |a₂| and |a₃| for the functions in these classes. Also we investigate upper bounds for the Fekete-Szegö functional and second Hankel determinant for these classes.