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Polynomials, sign patterns and Descartes' rule of signs

Vladimir Petrov Kostov (2019)

Mathematica Bohemica

By Descartes’ rule of signs, a real degree d polynomial P with all nonvanishing coefficients with c sign changes and p sign preservations in the sequence of its coefficients ( c + p = d ) has pos ≤ c positive and ¬ ≤ p negative roots, where pos ≡ c ( mod 2 ) and ¬ ≡ p ( mod 2 ) . For 1 ≤ d ≤ 3 , for every possible choice of the sequence of signs of coefficients of P (called sign pattern) and for every pair ( pos , neg ) satisfying these conditions there exists a polynomial P with exactly pos positive and exactly ¬ negative roots (all of them simple). For d ≥ 4 this is not...

Problems Concerning Subclasses of Analytic Functions

Maslina Darus, Imran Faisal (2012)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

In the present paper, we establish some interesting results concerning the quasi-Hadamard product for certain subclasses of analytic functions.

Properties of differences of meromorphic functions

Zong-Xuan Chen, Kwang Ho Shon (2011)

Czechoslovak Mathematical Journal

Let f be a transcendental meromorphic function. We propose a number of results concerning zeros and fixed points of the difference g ( z ) = f ( z + c ) - f ( z ) and the divided difference g ( z ) / f ( z ) .

Properties of functions concerned with Carathéodory functions

Mamoru Nunokawa, Emel Yavuz, Shigeyoshi Owa (2013)

Annales UMCS, Mathematica

Let Pn denote the class of analytic functions p(z) of the form p(z) = 1+cnzn + cn+1zn+1 + ... in the open unit disc U . Applying the result by S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65 (1978), 289-305), some interesting properties for p(z) concerned with Carath´eodory functions are discussed. Further, some corollaries of the results concerned with the result due to M. Obradović and S. Owa (Math. Nachr. 140 (1989), 97-102) are shown.

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