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Search: MSC category 26D07 ( Inequalities involving other types of functions )

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1. CMB 2016 (vol 59 pp. 225)

Atıcı, Ferhan M.; Yaldız, Hatice
Convex Functions on Discrete Time Domains
In this paper, we introduce the definition of a convex real valued function $f$ defined on the set of integers, ${\mathbb{Z}}$. We prove that $f$ is convex on ${\mathbb{Z}}$ if and only if $\Delta^{2}f \geq 0$ on ${\mathbb{Z}}$. As a first application of this new concept, we state and prove discrete Hermite-Hadamard inequality using the basics of discrete calculus (i.e. the calculus on ${\mathbb{Z}}$). Second, we state and prove the discrete fractional Hermite-Hadamard inequality using the basics of discrete fractional calculus. We close the paper by defining the convexity of a real valued function on any time scale.

Keywords:discrete calculus, discrete fractional calculus, convex functions, discrete Hermite-Hadamard inequality
Categories:26B25, 26A33, 39A12, 39A70, 26E70, 26D07, 26D10, 26D15

2. CMB 1999 (vol 42 pp. 417)

Aziz-Ul-Auzeem, Abdul; Zarger, B. A.
Some Properties of Rational Functions with Prescribed Poles
Let $P(z)$ be a polynomial of degree not exceeding $n$ and let $W(z) = \prod^n_{j=1}(z-a_j)$ where $|a_j|> 1$, $j =1,2,\dots, n$. If the rational function $r(z) = P(z)/W(z)$ does not vanish in $|z| < k$, then for $k=1$ it is known that $$ |r'(z)| \leq \frac{1}{2} |B'(z)| \Sup_{|z|=1} |r(z)| $$ where $B(Z) = W^\ast(z)/W(z)$ and $W^\ast (z) = z^n \overline {W(1/\bar z)}$. In the paper we consider the case when $k>1$ and obtain a sharp result. We also show that $$ \Sup_{|z|=1} \biggl\{ \biggl| \frac{r'(z)}{B'(z)} \biggr| +\biggr| \frac{\bigl(r^\ast (z)\bigr)'}{B'(z)} \biggr| \biggr\} = \Sup_{|z|=1} |r(z)| $$ where $r^\ast (z) = B(z) \overline{r(1/\bar z)}$, and as a consequence of this result, we present a generalization of a theorem of O'Hara and Rodriguez for self-inversive polynomials. Finally, we establish a similar result when supremum is replaced by infimum for a rational function which has all its zeros in the unit circle.


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