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An approach to the generalized Bessel–Maitland function is proposed in the present paper. It is denoted by$\mathcal{J}_{\nu , \lambda }^{\mu }$Jν,λμ, where$\mu >0$μ>0and$\lambda ,\nu \in \mathbb{C\ }$λ,ν∈Cget increasing interest from both theoretical mathematicians and applied scientists. The main objective is to establish the integral representation of$\mathcal{J}_{\nu ,\lambda }^{\mu }$Jν,λμby applying Gauss’s multiplication theorem and the representation for the beta function as well as Mellin–Barnes representation using the residue theorem. Moreover, themth derivative of$\mathcal{J}_{\nu ,\lambda }^{\mu }$Jν,λμis considered, and it turns out that it is expressed as the Fox–Wright function. In addition, the recurrence formulae and other identities involving the derivatives are derived. Finally, the monotonicity of the ratio between two modified Bessel–Maitland functions$\mathcal{I}_{\nu ,\lambda }^{\mu }$Iν,λμdefined by$\mathcal{I}_{\nu ,\lambda }^{\mu }(z)=i^{-2\lambda -\nu }\mathcal{J}_{ \nu ,\lambda }^{\mu }(iz)$Iν,λμ(z)=i−2λ−νJν,λμ(iz)of a different order, the ratio between modified Bessel–Maitland and hyperbolic functions, and some monotonicity results for$\mathcal{I}_{\nu ,\lambda }^{\mu }(z)$Iν,λμ(z)are obtained where the main idea of the proofs comes from the monotonicity of the quotient of two Maclaurin series. As an application, some inequalities (like Turán-type inequalities and their reverse) are proved. Further investigations on this function are underway and will be reported in a forthcoming paper.

The scope of our investigation is to study the geometric properties of the normalized form of the combination of generalized Lommel–Wright function $J_{\nu ,\lambda }^{\mu ,m}$ J ν , λ μ , m defined by $\mathfrak{J}_{\nu ,\lambda }^{\mu ,m}(z):=\Gamma ^{m}(\lambda +1) \Gamma (\lambda +\nu +1)2^{2\lambda +\nu } z^{1-(\nu /2)-\lambda } \mathcal{I}_{\nu ,\lambda }^{\mu ,m}(\sqrt{z})$ J ν , λ μ , m ( z ) : = Γ m ( λ + 1 ) Γ ( λ + ν + 1 ) 2 2 λ + ν z 1 − ( ν / 2 ) − λ I ν , λ μ , m ( z ) , where $\mathcal{I}_{\nu ,\lambda }^{\mu ,m}(z):=(1-2\lambda -\nu )J_{\nu , \lambda }^{\mu ,m}(z)+z (J_{\nu ,\lambda }^{\mu ,m}(z) )^{ \prime }$ I ν , λ μ , m ( z ) : = ( 1 − 2 λ − ν ) J ν , λ μ , m ( z ) + z ( J ν , λ μ , m ( z ) ) ′ and $$ J_{\nu ,\lambda }^{\mu ,m}(z)= \biggl(\frac{z}{2} \biggr)^{2\lambda + \nu } \sum_{n=0}^{\infty } \frac{(-1)^{n}}{\Gamma ^{m} (n+\lambda +1 )\Gamma (n\mu +\nu +\lambda +1 )} \biggl(\frac{z}{2} \biggr)^{2n}, $$ J ν , λ μ , m ( z ) = ( z 2 ) 2 λ + ν ∑ n = 0 ∞ ( − 1 ) n Γ m ( n + λ + 1 ) Γ ( n μ + ν + λ + 1 ) ( z 2 ) 2 n , with $m\in \mathbb{N}$ m ∈ N , $\mu >0$ μ > 0 and $\lambda ,\nu \in \mathbb{C}$ λ , ν ∈ C , including starlikeness and convexity of order α ($0\leq \alpha 0 ≤ α < 1 ) in the open unit disc using the two-sided inequality for the Fox–Wright functions that has been proved by Pogány and Srivastava in (Comput. Math. Appl. 57(1):127–140, 2009). Further, the orders of starlikeness and convexity are also evaluated using some classical tools. We then compare the orders of starlikeness and convexity given by both techniques to illustrate the efficacy of the approach. In addition, we proved that for some values of α, if $\lambda >-1$ λ > − 1 then $\operatorname{Re} (\mathfrak{J}_{\nu ,\lambda }^{\mu ,m}(z)/z )>\alpha $ Re ( J ν , λ μ , m ( z ) / z ) > α , $z\in \mathbb{U}$ z ∈ U , and if $\lambda \ge (\sqrt{10}-6 )/4$ λ ≥ ( 10 − 6 ) / 4 then the function $(\mathfrak{J}_{\nu ,\lambda }^{\mu ,m}(z^{2})/z )\ast \sin z$ ( J ν , λ μ , m ( z 2 ) / z ) ∗ sin z is close-to-convex with respect to $1/2\log ((1+z)/(1-z) )$ 1 / 2 log ( ( 1 + z ) / ( 1 − z ) ) where ∗ stands for the Hadamard product (or convolution) of two power series.

Using the third-order differential subordination basic results, we introduce certain classes of admissible functions and investigate some applications of third-order differential subordination for p-valent functions associated with generalized fractional differintegral operator.

The main object of this paper is to investigate some subordination results of certain subclasses of multivalent analytic functions which are defined by a generalized fractional differintegral operator. Inclusion relations for functions in the class $\mathcal {S}_{p,q}^{\lambda,\mu,\eta }\left (\zeta ;A,B\right) \ $ and the images of these functions by the generalized Bernardi-Libera-Livingston integral operator are also considered.

Making use of the operator L q , ν associated with functions of the form f ( z ) = 1 z + ∑ k = 1 ∞ a k z k − 1 , which are analytic in the punctured unit disc U * : = U ∖ { 0 } , we introduce two subclasses of meromorphic functions and investigate convolution properties and coefficient estimates for these subclasses.

The purpose of this paper is to introduce necessary and sufficient condition of (Gaussian) hypergeometric functions to be in a subclass of uniformly starlike and uniformly convex functions. Operators related to hypergeometric functions are also considered. Some of our results correct previously known results.