Spectral Analysis of a Class of Non-Self-Adjoint Differential Operator Pencils with a Generalized Function

2005 ◽  
Vol 145 (1) ◽  
pp. 1457-1461 ◽  
Author(s):  
R. F. Efendiev
Keyword(s):  
Spectral Analysis ◽  
Differential Operator ◽  
Generalized Function ◽  
Operator Pencils

scholarly journals Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator

2021 ◽  
Vol 19 (1) ◽  
pp. 329-337
Author(s):  
Huo Tang ◽  
Kaliappan Vijaya ◽  
Gangadharan Murugusundaramoorthy ◽  
Srikandan Sivasubramanian
Keyword(s):  
Analytic Function ◽  
Differential Operator ◽  
Lower Bounds ◽  
Analytic Functions ◽  
Function Class ◽  
Partial Sums ◽  
Generalized Function ◽  
Inclusion Relations ◽  
Alpha Beta

Abstract Let f k ( z ) = z + ∑ n = 2 k a n z n {f}_{k}\left(z)=z+{\sum }_{n=2}^{k}{a}_{n}{z}^{n} be the sequence of partial sums of the analytic function f ( z ) = z + ∑ n = 2 ∞ a n z n f\left(z)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n} . In this paper, we determine sharp lower bounds for Re { f ( z ) / f k ( z ) } {\rm{Re}}\{f\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}\left(z)\} , Re { f k ( z ) / f ( z ) } {\rm{Re}}\{{f}_{k}\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}f\left(z)\} , Re { f ′ ( z ) / f k ′ ( z ) } {\rm{Re}}\{{f}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}^{^{\prime} }\left(z)\} and Re { f k ′ ( z ) / f ′ ( z ) } {\rm{Re}}\{{f}_{k}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}^{^{\prime} }\left(z)\} , where f ( z ) f\left(z) belongs to the subclass J p , q m ( μ , α , β ) {{\mathcal{J}}}_{p,q}^{m}\left(\mu ,\alpha ,\beta ) of analytic functions, defined by Sălăgean ( p , q ) \left(p,q) -differential operator. In addition, the inclusion relations involving N δ ( e ) {N}_{\delta }\left(e) of this generalized function class are considered.

scholarly journals On Singular Dissipative Fourth-Order Differential Operator in Lim-4 Case

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Ekin Uğurlu ◽  
Elgiz Bairamov
Keyword(s):  
Spectral Analysis ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Boundary Value ◽  
Inverse Operator ◽  
Order Differential Operator ◽  
Associated Functions ◽  
Fourth Order Differential Operator

A singular dissipative fourth-order differential operator in lim-4 case is considered. To investigate the spectral analysis of this operator, it is passed to the inverse operator with the help of Everitt's method. Finally, using Lidskiĭ's theorem, it is proved that the system of all eigen- and associated functions of this operator (also the boundary value problem) is complete.

scholarly journals On a generalized function-to-sequence transform

2020 ◽  
pp. 5-5
Author(s):  
Slobodan Trickovic ◽  
Miomir Stankovic
Keyword(s):  
Differential Operator ◽  
Generalized Function ◽  
Discrete Operator ◽  
Borel Transform ◽  
Inverse Transform ◽  
Binomial Transform

By attaching a sequence {?n}n?N0 to the binomial transform, a new operator D? is obtained. We use the same sequence to define a new transform T? mapping derivatives to the powers of D?, and integrals to D-1?. The inverse transform B? of T? is introduced and its properties are studied. For ?n = (-1)n, B? reduces to the Borel transform. Applying T? to Bessel's differential operator d/dx x d/dx, we obtain Bessel's discrete operator D?nN?. Its eigenvectors correspond to eigenfunctions of Bessel's differential operator.

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