We will apply a fixed point method for proving the Hyers-Ulam stability of the functional equationf(x +y) = f(x)f(y) f(x)+f(y) .

/pdf/the-australian-journal-of-mathematical-analysis-and-3ja4vnapvl.pdf

The Australian Journal of Mathematical Analysis and Applications

/pdf/l-q-norm-inequalities-for-the-polar-derivative-of-a-13wvl2sk8q.pdf

L_q Norm inequalities for the polar derivative of a polynomial

For a polynomial p(z) of degree n, we consider an operator Dα which map a polynomial p(z) into Dαp(z) := (α-z)p'(z)+ np(z) with respect to α. It was proved by Liman et al. [A. Liman, R. N. Mohapatra and W. M. Shah, Inequalities for the Polar Derivative of a Polynomial, Complex Analysis and Operator Theory, 2010] that if p(z) has no zeros in |z| < 1 then for all α, β _ C with |α| ≥ 1, |β| ≤ 1 and |z| = 1, |zDαp(z) + nβ |α| 1 2 p(z) |≤ n 2 {[| α+β |α| 1 2 |] + max | p (z)|z|-1 |} - [|α + β |α| - 1 2 |-| z + β |α| - 1 2|] min | p(z) |z|=1|}. In this paper we extend above inequality for the polynomials having no zeros in |z| < 1, except s-fold zeros at the origin. Our result generalize certain well-known polynomial inequalities.

/pdf/inequalities-for-the-polar-derivative-of-a-polynomial-with-1uiul163s6.pdf

Inequalities for the polar derivative of a polynomial with restricted zeros

Let P(z) be a polynomial of degree n. We consider an operator D
 𝛼 that maps P(z) into D
 𝛼
 P(z) := nP(z) + (𝛼 − z)P0(z) and establish some results concerning the estimates of |D
 𝛼
 P(z)| in the disk |z| = R ≥ 1, and thereby obtain extensions and generalizations of numerous well-known inequalities for polynomial.

On an Operator Preserving Inequalities Between Polynomials

If P(z) is a polynomial of degree n, and α a complex number, then polar derivative of P(z) with respect to the point α, denoted by D α P(z), is defined by 

$$\displaystyle{D_{\alpha }P(z) = nP(z) + (\alpha -z)P^{{\prime}}(z).}$$

Clearly, D α P(z) is a polynomial of degree n − 1, and it generalizes the ordinary derivative in the sense that 

$$\displaystyle{\lim _{\alpha \rightarrow \infty }\left [\frac{D_{\alpha }P(z)} {\alpha } \right ] = P^{{\prime}}(z).}$$

It is well known that if P(z) is a polynomial of degree n, then max | z | = 1 | P ′ (z) | ≤ nmax | z | = 1 | P(z) | . This inequality is known as Bernstein’s inequality (Bernstein, Lecons sur les proprietes extremales et la meilleure approximation des fonctions analytiques d’une variable reelle. Gauthier-Villars, Paris, 1926), although this inequality was also proved by Riesz (Jahresber Dtsch Math-Verein 23:354–368, 1914) about 12 years before it was proved by Bernstein. The subject of inequalities for polynomials and related classes of functions plays an important and crucial role in obtaining inverse theorems in Approximation Theory. Frequently, the further progress in inverse theorems has depended upon first obtaining the corresponding analogue or generalization of Markov’s and Bernstein’s inequalities. These inequalities have been the starting point of a considerable literature in Mathematics, and was one of the areas in which Professor Q. I. Rahman worked for more than 50 years, and made some of the most important and significant contributions. Over a period, this Bernstein’s inequality and corresponding inequality concerning the growth of polynomials have been generalized in different domains, in different norms, and for different classes of functions, and in the literature one can find hundreds of papers on this topic. Here we study some of the research centered around Bernstein-type inequalities for polar derivatives of polynomials. The chapter is purely expository in nature and an attempt has been made here to provide results starting from the beginning to some of the most recent ones.

On Bernstein-Type Inequalities for the Polar Derivative of a Polynomial

Generalization of certain well-known polynomial inequalities

If P(z) is a polynomial of degree n, having all its zeros in the diskjzj k, k 1, then it was shown by Govil (Proc. Amer. Math. Soc. 41, no. 2 (1973), 543-546) that maxzj=1 P 0 (z) n 1 +kn max jzj=1 jP(z)j: In this paper, we obtain generalization as well as improvement of above in- equality for the polynomial of the type P(z) = c0 + P n = c z , 1 n. Also we generalize a result due to Dewan and Mir (Southeast Asian Bull. Math. 31 (2007), 691-695) in this direction.

/pdf/on-maximum-modulus-for-the-derivative-of-a-polynomial-2a77t502d7.pdf

On maximum modulus for the derivative of a polynomial

/pdf/growth-of-polynomials-whose-zeros-are-outside-a-circle-556ntcq5qd.pdf

Growth of polynomials whose zeros are outside a circle

If is a polynomial of degree n having all its zeros in |z|≤ k , k ≥ 1 , then Govil [Proc Amer Math Soc, Vol 41 (1973), pp 543–546] proved In this paper, we generalize and extend the above inequality and some other related inequalities by restricting ourselves to polynomials of degree n having all its zeros in |z|≤ · k , k ≥ 1 with s -fold zeros at the origin where 0 ≤ s ≤ n 

Turán-type inequalities for polynomials with restricted zeros

In this paper we obtain certain results for the polar derivative of a polynomial $p(z) = c_nz^n +\sum_{j=\mu}^n c_{n-j}z^{n-j}$, $1\leq\mu\leq n$, having all its zeros on $|z| = k$, $k\leq 1$, which generalizes the results due to Dewan and Mir, Dewan and Hans We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros [Editor's note: There are flaws in the paper, see M A Qazi, Remarks on some recent results about polynomials with restricted zeros, Ann Univ Mariae Curie-Sklodowska Sect A 67 (2), (2013), 59-64 ]

/pdf/inequalities-concerning-polar-derivative-of-polynomials-4gcjvw65jb.pdf

Inequalities concerning polar derivative of polynomials

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