International Journal of Advances in Mathematics
International Journal of Advances in Mathematics
2456-6098
Volume 2018
Number 2
2018
March
1
On a Conjecture on Unimodal Sequences
9
16
EN
Gholamhassan
Shirdel
Faculty of Science, Department of Mathematics\\
University of Qom, Qom. Iran.
shirdel81math@gmail.com
Maryam
Sarabadan
Faculty of Science, Department of Mathematics\\
University of Qom, Qom. Iran.
shirdel81math@gmail.com
Wang and Yeh proved that if $P(x)$ is a polynomial with non-
negative and nondecreasing coefficients, then $P(x + d)$ is unimodal for any
$d > 0$. A mode of a unimodal polynomial $f(x) = a_0 + a_1x +\cdots+a_mx^m$ is
an index $k$ such that $a_k$ is the maximum coefficient. Suppose that $M_*(P, d)$
is the smallest mode of $P(x + d)$, and $M^*(P, d)$ the greatest mode. Wang
and Yeh conjectured that if $d_2 > d_1 > 0$, then $M_*(P, d_1)\geq M_*(P, d_2)$ and
$M^*(P, d_1) \geq M^*(P, d_2)$. This conjecture has already been proved in [7] but we give a different proof of this conjecture.
We also show that if $\{d_j:\,\,0\leq j\leq m\}$ is a unimodal sequence, then there is a polynomial $p(x)=\sum_{i=0}^m a_ix^i$ with nonnegative and non- decreasing coefficients such that $p(x+n)=\sum_{j=0}^m d_jx^j$, where $n$ is a positive integer number. Furthermore, we define the almost unimodal sequences and prove that under some conditions the polynomial $p(x^k+d)$ for any positive real number $d$ and integer number $k\geq2$, is almost unimodal.
Almost unimodality; unimodality; unimodal polynomial;
mode.
http://adv-math.com/conjecture-unimodal-sequences/
http://adv-math.com/wp-content/uploads/2018/03/ADV-201742.pdf