Year 2019,
, 87 - 101, 01.02.2019
P. Özkartepe
,
F.g. Abdullayev
References
- F.G. Abdullayev, On the some properties on orthogonal polynomials over the regions
of complex plane 1. Ukr. Math. J. 52 (12), 1807-1817, 2000.
- F.G. Abdullayev, On the interference of the weight boundary contour orthogonal polynomials
over the region, J. of Comp. Anal. and Appl. 6 (1), 31-42, 2004.
- F.G. Abdullayev and V.V. Andrievskii, On the orthogonal polynomials in the domains
with $K$-quasiconformal boundary, Izv. Akad. Nauk Azerb. SSR., Ser. FTM, 1, 3-7,
1983.
- F.G. Abdullayev and C.D. Gün, On the behavior of the algebraic polynomials in
regions with piecewise smooth boundary without cusps, Ann. Polon. Math. 111, 39-58,
2014.
- F.G. Abdullayev and N.P. Özkartepe, On the Behavior of the Algebraic Polynomial
in Unbounded Regions with Piecewise Dini -Smooth Boundary, Ukr. Math. J., 66 (5),
579-597, 2014.
- F.G. Abdullayev and N.P. Özkartepe, On the growth of algebraic polynomials in the
whole complex plane, J. Korean Math. Soc. 52 (4) 699-725, 2015.
- F.G. Abdullayev and N.P. Özkartepe, Uniform and pointwise polynomial inequalities
in regions with cusps in the weighted Lebesgue space, Jaen J. Approx. 7 (2), 231-261,
2015.
- F.G. Abdullayev, N.P. Özkartepe, C.D. Gün, Uniform and pointwise polynomial inequalities
in regions without cusps in the weighted Lebesgue space, Bulletin of Tbilisi
ICMC 18 (1), 146-167, 2014.
- L. Ahlfors, Lectures on Quasiconformal Mappings, Princeton, NJ: Van Nostrand,
1966.
- V.V. Andrievskii, Weighted Polynomial Inequalities in the Complex Plane, J. Approx.
Theory 164(9), 1165-1183, 2012.
- V.V. Andrievskii, V.I. Belyi, and V.K. Dzyadyk, Conformal invariants in constructive
theory of functions of complex plane, Atlanta: World Federation Publ.Com., 1995.
- E. Hille, G. Szegö, and J.D. Tamarkin, On some generalization of a theorem of A.
Markoff, Duke Math. 3, 729-739, 1937.
- D. Jackson, Certain problems on closest approximations, Bull. Amer. Math. Soc. 39,
889-906, 1933.
- O. Lehto and K.I. Virtanen, Quasiconformal Mapping in the Plane, Springer Verlag,
Berlin, 1973.
- F.D. Lesley, Hölder continuity of conformal mappings at the boundary via the strip
method, Indiana Univ. Math. J. 31, 341-354, 1982.
- D.I. Mamedhanov, Inequalities of S.M.Nikol’skii type for polynomials in the complex
variable on curves, Soviet Math. Dokl. 15, 34-37, 1974.
- D.I. Mamedhanov, On Nikol’skii-type inequalities with new characteristic, Dokl.
Math. 82, 882-883, 2010.
- G.V. Milovanovic, D.S. Mitrinovic, and Th.M. Rassias, Topics in Polynomials: Extremal
Problems, Inequalities, Zeros, World Scientific, Singapore, 1994.
- S.M. Nikol’skii, Approximation of Function of Several Variable and Imbeding Theorems,
Springer-Verlag, New-York, 1975.
- N.P. Özkartepe and F.G. Abdullayev, Interference of the weight and boundary contour
for algebraic polynomials in the weighted Lebesgue spaces I, Ukr. Math. J. 68(10),
1574-1590, 2017.
- Ch. Pommerenke, Univalent Functions, Göttingen, Vandenhoeck & Ruprecht, 1975.
- Ch. Pommerenke, Boundary Behavior of Conformal Maps, Springer-Verlag, Berlin,
1992.
- I. Pritsker, Comparing Norms of Polynomials in One and Several Variables, J. Math.
Anal. Appl. 216, 685-695, 1997.
- S. Rickman, Characterisation of quasiconformal arcs, Ann. Acad. Sci. Fenn., Ser. A,
Mathematica 395, 30 p, 1966.
- G. Szegö and A. Zigmund , On certain mean values of polynomials, J. Anal. Math.
3, 225-244, 1954.
- P.K. Suetin, The ordinally comparison of various norms of polynomials in the complex
domain, Matematicheskie zapiski Uralskogo Gos. Universiteta 5 (4), 1966 (in
Russian).
- P.K. Suetin, Main properties of the orthogonal polynomials along a circle, Uspekhi
Math. Nauk 21 (2 (128)), 41-88, 1966.
- P.K. Suetin, On some estimates of the orthogonal polynomials with singularities weight
and contour, Sib. Math. J VIII (3), 1070-1078, 1967 (in Russian).
- J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex
Domain, AMS, 1960.
- S.E. Warschawski, On differentiability at the boundary in conformal mapping, Proc.
Amer. Math. Soc., 12, 614-620, 1961.
The uniform and pointwise estimates for polynomials on the weighted Lebesgue spaces in the general regions of complex plane
Year 2019,
, 87 - 101, 01.02.2019
P. Özkartepe
,
F.g. Abdullayev
Abstract
The estimation of the modulus of algebraic polynomials on the boundary contour with weight function, having some singularities, with respect to the their quasinorm, on the weighted Lebesgue space was studied in this current work.
References
- F.G. Abdullayev, On the some properties on orthogonal polynomials over the regions
of complex plane 1. Ukr. Math. J. 52 (12), 1807-1817, 2000.
- F.G. Abdullayev, On the interference of the weight boundary contour orthogonal polynomials
over the region, J. of Comp. Anal. and Appl. 6 (1), 31-42, 2004.
- F.G. Abdullayev and V.V. Andrievskii, On the orthogonal polynomials in the domains
with $K$-quasiconformal boundary, Izv. Akad. Nauk Azerb. SSR., Ser. FTM, 1, 3-7,
1983.
- F.G. Abdullayev and C.D. Gün, On the behavior of the algebraic polynomials in
regions with piecewise smooth boundary without cusps, Ann. Polon. Math. 111, 39-58,
2014.
- F.G. Abdullayev and N.P. Özkartepe, On the Behavior of the Algebraic Polynomial
in Unbounded Regions with Piecewise Dini -Smooth Boundary, Ukr. Math. J., 66 (5),
579-597, 2014.
- F.G. Abdullayev and N.P. Özkartepe, On the growth of algebraic polynomials in the
whole complex plane, J. Korean Math. Soc. 52 (4) 699-725, 2015.
- F.G. Abdullayev and N.P. Özkartepe, Uniform and pointwise polynomial inequalities
in regions with cusps in the weighted Lebesgue space, Jaen J. Approx. 7 (2), 231-261,
2015.
- F.G. Abdullayev, N.P. Özkartepe, C.D. Gün, Uniform and pointwise polynomial inequalities
in regions without cusps in the weighted Lebesgue space, Bulletin of Tbilisi
ICMC 18 (1), 146-167, 2014.
- L. Ahlfors, Lectures on Quasiconformal Mappings, Princeton, NJ: Van Nostrand,
1966.
- V.V. Andrievskii, Weighted Polynomial Inequalities in the Complex Plane, J. Approx.
Theory 164(9), 1165-1183, 2012.
- V.V. Andrievskii, V.I. Belyi, and V.K. Dzyadyk, Conformal invariants in constructive
theory of functions of complex plane, Atlanta: World Federation Publ.Com., 1995.
- E. Hille, G. Szegö, and J.D. Tamarkin, On some generalization of a theorem of A.
Markoff, Duke Math. 3, 729-739, 1937.
- D. Jackson, Certain problems on closest approximations, Bull. Amer. Math. Soc. 39,
889-906, 1933.
- O. Lehto and K.I. Virtanen, Quasiconformal Mapping in the Plane, Springer Verlag,
Berlin, 1973.
- F.D. Lesley, Hölder continuity of conformal mappings at the boundary via the strip
method, Indiana Univ. Math. J. 31, 341-354, 1982.
- D.I. Mamedhanov, Inequalities of S.M.Nikol’skii type for polynomials in the complex
variable on curves, Soviet Math. Dokl. 15, 34-37, 1974.
- D.I. Mamedhanov, On Nikol’skii-type inequalities with new characteristic, Dokl.
Math. 82, 882-883, 2010.
- G.V. Milovanovic, D.S. Mitrinovic, and Th.M. Rassias, Topics in Polynomials: Extremal
Problems, Inequalities, Zeros, World Scientific, Singapore, 1994.
- S.M. Nikol’skii, Approximation of Function of Several Variable and Imbeding Theorems,
Springer-Verlag, New-York, 1975.
- N.P. Özkartepe and F.G. Abdullayev, Interference of the weight and boundary contour
for algebraic polynomials in the weighted Lebesgue spaces I, Ukr. Math. J. 68(10),
1574-1590, 2017.
- Ch. Pommerenke, Univalent Functions, Göttingen, Vandenhoeck & Ruprecht, 1975.
- Ch. Pommerenke, Boundary Behavior of Conformal Maps, Springer-Verlag, Berlin,
1992.
- I. Pritsker, Comparing Norms of Polynomials in One and Several Variables, J. Math.
Anal. Appl. 216, 685-695, 1997.
- S. Rickman, Characterisation of quasiconformal arcs, Ann. Acad. Sci. Fenn., Ser. A,
Mathematica 395, 30 p, 1966.
- G. Szegö and A. Zigmund , On certain mean values of polynomials, J. Anal. Math.
3, 225-244, 1954.
- P.K. Suetin, The ordinally comparison of various norms of polynomials in the complex
domain, Matematicheskie zapiski Uralskogo Gos. Universiteta 5 (4), 1966 (in
Russian).
- P.K. Suetin, Main properties of the orthogonal polynomials along a circle, Uspekhi
Math. Nauk 21 (2 (128)), 41-88, 1966.
- P.K. Suetin, On some estimates of the orthogonal polynomials with singularities weight
and contour, Sib. Math. J VIII (3), 1070-1078, 1967 (in Russian).
- J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex
Domain, AMS, 1960.
- S.E. Warschawski, On differentiability at the boundary in conformal mapping, Proc.
Amer. Math. Soc., 12, 614-620, 1961.