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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 ( J 1 λ , μ f ( z ) ) ' ( 1 - γ ) J 1 λ , μ f ( z ) + γ z 2 ( J 1 λ , μ f ( 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 A γ λ , 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 A γ , c m λ , 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 A γ , c m λ , μ , ν ( 1 , β ) A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right) .

Indulal and Balakrishnan (2016) have put forward the Indu-Bala product and based on this product four new operations are defined by the authors of this manuscript in the paper “Four new operations of graphs based on Indu-Bala product and the Zagreb indices”. In this paper we establish explicit formulas of the forgotten topological index and reduced second Zagreb index in connection with these new operations of graphs.

In this paper we obtain some bounds in terms of polynomials for the function sin x x {{\sin x} \over x} , x ∈ [0, π].

In the current paper, we introduce a new family for holomorphic functions defined by Wanas operator associated with Poisson distribution series. Also we derive some interesting geometric properties for functions belongs to this family.

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.

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 𝒟 γ ( f , g ) : = 1 w - u ∫ γ f ( z ) g ( z ) d z - 1 w - u ∫ γ f ( z ) d z 1 w - u ∫ γ g ( z ) d z . {{\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 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 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 ∫ a c f ( x ) d x + ( c - a ) g ( c ) = ∫ c b g ( x ) d x + ( 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.

In the present paper we establish some theorems on the existence of common approximate fixed points for a pair of generalized contractive type mappings with the property that the diameter of the set of common ∈−fixed points is tending to zero as ∈ tends to zero.

In this paper we obtain estimations of the errors in approximation by positive linear operators which fix certain functions. We use both the first and the second order classical moduli of smoothness and a generalized modulus of continuity of order two. Some applications involving Bernstein type operators, Kantorovich type operators and genuine Bernstein-Durrmeyer type operators are presented.

In this paper we have introduced two new types of sets termed as 𝕀* µ sets and strongly 𝕀* µ -open sets and discussed some of its properties. The relation between similar types of sets, characterizations and some basic properties of such sets have been studied.

In this paper an improved error bound is obtained for the complete quartic spline with deficiency 2, in the less smooth class of continuous functions. In the case of Lipschitzian functions, the obtained estimate improves the constant from Theorem 3, in J. Approx. Theory 58 (1989) 58-67. Some applications of the complete quartic spline in the numerical integration and in the construction of an iterative numerical method for fourth order two-point boundary value problems with pantograph type delay are presented.

In this paper we consider two continuous functions f, g : [a, b] → ℝ and we study for these ones, under which circumstances the intermediate point function is four order di erentiable at the point x = a and we calculate its derivative.

In this paper, we find accordance of some classical inequalities and fractional dynamic inequalities. We find inequalities such as Radon’s inequality, Bergström’s inequality, Rogers-Hölder’s inequality, Cauchy-Schwarz’s inequality, the weighted power mean inequality and Schlömilch’s inequality in generalized and extended form by using the Riemann-Liouville fractional integrals on time scales.

This paper mainly dedicated on overview of zero sets in Ideal topological spaces. We also introduce a new class of functions which generalizes the class of continuous functions and investigate its position in the hierarchy of continuous functions on Ideal topological spaces. Moreover, these new sets (zero*-ℐ-set) which is a pragmatic approach to characterize completely Hausdorff spaces.

In this paper, we investigate the existence of solution of integro-differential equations (IDEs) with Hilfer-Hadamard fractional derivative. The main results are obtained by using Schaefer’s fixed point theorem. Some Ulam stability results are presented.