Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCertain identities on derivatives of radial homogeneous and logarithmic functions
96
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $k$ be a natural number and $s$ be real. In the 1-dimensional case, the $k$-th order derivatives of the functions $lvert xrvert^s$ and $log lvert xrvert$ are multiples of $lvert xrvert^{s-k}$ and $lvert xrvert^{-k}$, respectively. In the present paper, we generalize this fact to higher dimensions by introducing a suitable norm of the derivatives, and give the exact values of the multiples.
rate research
Read More
Applying the theory of elliptic functions we establish two Jacobi theta function identities. From these identities we confirm two q-trigonometric identities conjectured by Gosper. As an application, we give a new and simple proof of a Pi_{q}-identity of Gosper.
In this paper necessary and sufficient conditions are deduced for the starlikeness of Bessel functions of the first kind and their derivatives of the second and third order by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some Mittag-Leffler expansions for the derivatives of Bessel functions of the first kind, as well as some results on the zeros of these functions.
In this paper, sums represented in (3) are studied. The expressions are derived in terms of Bessel functions of the first and second kinds and their integrals. Further, we point out the integrals can be written as a Meijer G function.
We establish a characterization of the Hardy spaces on the homogeneous groups in terms of the Littlewood-Paley functions. The proof is based on vector-valued inequalities shown by applying the Peetre maximal function.
It is well known that a continuously differentiable function is monotone in an interval $[a,b]$ if and only if its first derivative does not change its sign there. We prove that this is equivalent to requiring that the Caputo derivatives of all orders $alpha in (0,1)$ with starting point $a$ of this function do not have a change of sign there. In contrast to what is occasionally conjectured, it not sufficient if the Caputo derivatives have a constant sign for a few values of $alpha in (0,1)$ only.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
