# Mathematica Moravica, Vol. 24, No. 1 (2020)

Stability and boundedness of nonautonomous neutral differential equation with delay
Mathematica Moravica, Vol. 24, No. 1 (2020), 1–16.

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Abstract. We consider the nonautonomous neutral differential equation with delay \begin{gather*} &\left[ p(t)\Big(q(t)\big(x(t)+\beta_{1} x(t-r_{1})\big)'\Big)'\right] ^{ \prime }+a(t)\big( x^{\prime \prime}(t)+ \beta_{2} x''(t-r_{2})\big)\\ &+b(t)\big(x^{\prime }(t)+\beta_{3} x'(t-r_{3})\big)+c(t)f(x(t-\sigma))=e(t, x, x', x''). \end{gather*} Using the method of Lyapunov, we give conditions for the uniform asymptotic stability and uniform boundedness and square integrability of solutions for the considered system. Our theorems generalize and extend some related results known in the literature. Example is given to show our results.
Keywords. Uniform ultimate boundedness, square integrability, Lyapunov functional, neutral differential equation of third order.

Differential sandwich results for Wanas operator of analytic functions
Mathematica Moravica, Vol. 24, No. 1 (2020), 17–28.

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Abstract. In the present article, we determine some subordination and superordination results involving Wanas operator for certain normalized analytic functions defined in the unit disk $\mathbb{U}$. These results are applied to establish sandwich results. Our results extend corresponding previously known results.
Keywords. Analytic function, differential subordination, differential superoordination, dominant, subordinant, Wanas operator.

Some common fixed point theorems on partial metric spaces satisfying implicit relation
Mathematica Moravica, Vol. 24, No. 1 (2020), 29–43.

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Abstract. The aim of this article is to prove some fixed point and common fixed point theorems on partial metric spaces satisfying implicit relation. Our results extend and generalize several results from the existing literature.
Keywords. Fixed point, common fixed point, implicit relation, partial metric space.

On exponentially $(h_{1},h_{2})$-convex functions and fractional integral inequalities related
Mathematica Moravica, Vol. 24, No. 1 (2020), 45–62.

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Abstract. In this work the concept of exponentially $(h_{1},h_{2})$-convex function is introduced and using it, the Hermite-Hadamard inequality and some bounds for the right side of this inequality, via Raina's fractional integral operator and generalized convex functions, are established.
Keywords. Exponentially $(h_{1},h_{2})$-convex function, Raina's fractional integral operator, fractional integral inequalities.

On $G$-transitive version of perfectly meager sets
Mathematica Moravica, Vol. 24, No. 1 (2020), 63–70.

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Abstract. We study the $G-$ invariant version of perfectly meager sets (a generalization of the notion of $\mathrm{AFC}'$ sets). We find the necessary and sufficient conditions for the inclusion $\mathrm{AFC}'_{G} \subseteq \mathcal{I}$. In particular, we partially characterize for which groups $G$ of automorphisms of the Cantor space every $\mathrm{AFC}'_{G}$ set is Lebesgue null.
Keywords. Perfectly meager sets, strongly meager sets, $\mathrm{AFC}'$ sets.

Some inequalities for Heinz operator mean
Mathematica Moravica, Vol. 24, No. 1 (2020), 71–82.

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Abstract. In this paper we obtain some new inequalities for Heinz operator mean.
Keywords. Young's Inequality, Convex functions, Arithmetic mean-Geometric mean inequality, Heinz means.

On relationships between $q$-product identities and combinatorial partition identities
Mathematica Moravica, Vol. 24, No. 1 (2020), 83–91.

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Abstract. Andrews et al. [2] discussed about the combinatorial partition identities. We aim to present some relationships between $q$-product identities and combinatorial partition identities, by using and combining known formulas.
Keywords. Combinatorial partition identities, Jacobi’s triple-product identity, Ramanujan’s theta functions, $q$-Product identities, Combinatorial partition identities, Euler’s Pentagonal Number Theorem, Rogers-Ramanujan continued fraction, Rogers-Ramanujan identities.

Inverse-$C$-class function on weak semi compatibility and fixed point theorems for expansive mappings in $G$-metric spaces
Mathematica Moravica, Vol. 24, No. 1 (2020), 93–108.

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Abstract. In this paper we introduce the concept of inverse $C$-class function in $G$-metric setting and established some fixed point theorems. We also put some examples in support of proved fixed point results.
Keywords. Common fixed point, inverse-$C$-class function.

Existence and Ulam stability results for nonlinear hybrid implicit Caputo fractional differential equations
Mathematica Moravica, Vol. 24, No. 1 (2020), 109–122.

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Abstract. In this paper, we study the existence, uniqueness and estimate of solutions for nonlinear hybrid implicit Caputo fractional differential equations by using the contraction mapping principle and the generalization of Gronwall’s inequality. After that, we also establish the Ulam stability for the problem at hand. Finally, an example is given to illustrate this work.
Keywords. Implicit fractional differential equations, Caputo fractional derivatives, fixed point theorems, existence, uniqueness, Ulam stability.