The generalization of the generalized Al-Oboudi differential operator

In this paper, firstly we define the generalization of the generalized Al-Oboudi differential operator. Then we also define new classes of analytic and p-valently starlike and convex functions with complex order by means of this new general differential operator. Our main purpose is to determine coefficient bounds for functions in certain subclasses of this classes, which are introduced here by means of a family of nonhomogeneous Cauchy-Euler differential equations. Relevant connections of some of the results obtained with those in earlier works are also provided.     

Certain subclasses of p-valently analytic functions involving a generalized fractional differintegral operator

By making use of the principle of subordination between analytic functions and the generalized fractional differintegral operator, we introduce and investigate some new subclasses of p-valently analytic functions in the open unit disk. Such results as inclusion relationships, integral-preserving properties, convolution properties, subordination and superordination properties, and sandwich theorems for these classes are derived.     

Some families of p-valently analytic functions established by fractional derivatives and their certain consequences

The main object of this investigation is to reveal some relations between certain families of p-valently analytic functions established by fractional calculus, and to point their various consequences out.     

Ordinary differential operator and some of its applications to certain meromorphically p-valent functions

By making use of the well-known assertions given in Miller and Mocanu (1978) [13] and Nunokawa (1993) [14], certain theorems concerning p-valently meromorphic (strongly) starlike and (strongly) convex functions obtained in this investigation are firstly proved and then their certain consequences which will be interesting or important for analytic and geometric function theory are pointed out.     

Generalization of certain subclasses of multivalent functions with negative coefficients defined by using a differential operator

Making use of a differential operator, we introduce and study a certain class SC_n(j,p,q,α,λ) of p-valently analytic functions with negative coefficients. In this paper, we obtain numerous sharp results including (for example ) coefficient estimates, distortion theorem, radii of close-to convexity, starlikeness and convexity and modified Hadamard products of functions belonging to the class SC_n(j,p,q,α,λ). Finally, several applications investigate an integral operator, and certain fractional calculus operators are also considered.     

On polyharmonic interpolation

In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from C^∞ or analytic functions in the ball BR. We prove two main results on the interpolation of C^∞ or analytic functions f in the ball BR by polyharmonic functions h of a given order of polyharmonicity p.     

Some subclasses of multivalent functions involving a certain linear operator

The authors investigate various inclusion and other properties of several subclasses of the class Ap of normalized p-valent analytic functions in the open unit disk, which are defined here by means of a certain linear operator. Problems involving generalized neighborhoods of analytic functions in the class Ap are investigated. Finally, some applications of fractional calculus operators are considered.     

Starlikeness and subordination properties of a linear operator

In this paper, we introduce a new class of p-valent analytic functions defined by using a linear operator L_k^α. For functions in this class H_k^α(p,λh) we estimate the coefficients. Furthermore, some subordination properties related to the operator L_k^α are also derived.     

Subordination properties of multivalent functions involving an extended fractional differintegral operator

The object of this article is to investigate inclusion, radius, and other various properties of subclasses of multivalent analytic functions, which are defined by using an extended version of the Owa-Srivastava fractional differintegral operator Ω(λ, p).     

On some classes of p-valent functions involving Carlson-Shaffer operator

Let , be a class of functions analytic in the open unit disc E. We use Carlson-Shaffer operator for p-valent functions to define and study certain classes of analytic functions. Inclusion results, a radius problem and some other interesting properties are discussed.     

New explicit solutions to the p-Laplace equation based on isoparametric foliations

In contrast to an infinite family of explicit examples of two-dimensional p-harmonic functions obtained by G. Aronsson in the late 80s, there is very little known about the higher-dimensional case. In this paper, we show how to use isoparametric polynomials to produce diverse examples of p-harmonic and biharmonic functions. Remarkably, for some distinguished values of p and the ambient dimension n this yields first examples of rational and algebraic p-harmonic functions. Moreover, we show that there are no p-harmonic polynomials of the isoparametric type. This supports a negative answer to a question proposed in 1980 by J. Lewis.     

On a transplant operator and explicit construction of Cauchy-type integral representations for p-analytic functions

We present a new technique for explicit construction of Cauchy kernels and Cauchy integral representations for a class of generalized analytic functions and p-analytic functions.     

On a Class of p- Valently α-Convex Functions

The object of the present is to prove some properties of a class M_p(α) of p-valently α-convex Functions in the unit disk. Also, an integral representation for functions belonging to the class M_p(α) is shown.     

Certain subclasses of p-valent functions involving the Dziok-Srivastava operator

Using the methods of differential subordination, we investigate some inclusion relationships of certain subclasses of analytic and p-valent functions which are defined by means of Dziok-Srivastava operator.     

On p-Valently Close-To-Convex and Starlike Functions

The object of the present paper is to prove some interesting sufficient conditions for p-valently close-to-convex and starlike functions in the unit disk.     

On the evaluation of some Selberg-like integrals

Several methods of evaluation are presented for a family {In,d,p} of Selberg-like integrals that arise in the computation of the algebraic-geometric degrees of a family of spherical nilpotent orbits associated to the symmetric space of a simple real Lie group. Adapting the technique of Nishiyama, Ochiai and Zhu, we present an explicit evaluation in terms of certain iterated sums over permutation groups. The resulting formula, however, is only valid when the integrand involves an even power of the Vandermonde determinant. We then apply, to the general case, the theory of symmetric functions and obtain an evaluation of the integral In,d,p as a product of polynomial of fixed degree times a particular product of gamma factors; thereby identifying the asymptotics of the integrals with respect to their parameters. Lastly, we derive a recursive formula for evaluation of another general class of Selberg-like integrals, by applying some of the technology of generalized hypergeometric functions.     

On Finsler transnormal functions

In this note we discuss a few properties of transnormal Finsler functions, i.e., the natural generalization of distance functions and isoparametric Finsler functions. In particular, we prove that critical level sets of an analytic transnormal function are submanifolds, and the partition of M into level sets is a Finsler partition, when the function is defined on a compact analytic manifold M.     

A study of a special family of analytic functions at infinity

In the present work we introduce certain new subclasses of analytic functions in the open unit disk U. Then by using a quasi-Hadamard product we discuss their applications in the space of analytic functions.     

Subclasses of bi-univalent functions defined by convolution

In this paper, we introduced two new subclasses of the function class Σ of bi-univalent functions analytic in the open unit disc defined by convolution. Furthermore, we find estimates on the coefficients ∣a₂∣ and ∣a₃∣ for functions in these new subclasses.     

Enumeration of (p,q)-parking functions

Parking functions are central in many aspects of combinatorics. We define in this communication a generalization of parking functions which we call (p₁,…,p_k)-parking functions. We give a characterization of them in terms of parking functions and we show that they can be interpreted as recurrent configurations in the sandpile model for some graphs. We also establish a correspondence with a Lukasiewicz language, which enables to enumerate (p₁,…,p_k)-parking functions as well as increasing ones.