Operator Inequalities Involved Wiener–Hopf Problems in the Open Unit Disk

  • Rabha W. Ibrahim
Part of the Springer Optimization and Its Applications book series (SOIA, volume 151)


In this effort, we employ some of the linear differential inequalities to achieve integral inequalities of the type Wiener–Hopf problems (WHP). We utilize the concept of subordination and its applications to gain linear integral operators in the open unit disk that preserve two classes of analytic functions with a positive real part. Linear second-order differential inequalities play a significant role in the field of complex differential equations. Our study is based on a neighborhood containing the origin. Therefore, the Wiener–Hopf problem is decomposed around the origin in the open unit disk using two different classes of analytic functions. Moreover, we suggest a generalization for WHP by utilizing some classes of entire functions. Special cases are given in the sequel as well. A necessary and sufficient condition for WHP to be averaging operator on a convex domain (in the open unit disk) is given by employing the subordination relation (inequality).


  1. 1.
    N. Wiener, Tauberian theorems. Ann. Math. 33, 1–100 (1932)MathSciNetCrossRefGoogle Scholar
  2. 2.
    J.B. Lawrie, I.D. Abrahams, A brief historical perspective of the Wiener–Hopf technique. J. Eng. Math. 59(4), 351–358 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    F.B. Vicente et al., Wiener–Hopf optimal control of a hydraulic canal prototype with fractional order dynamics. ISA Trans. 82, 130–144 (2018)CrossRefGoogle Scholar
  4. 4.
    B.P. Lampe, E.N. Rosenwasser, H2-optimization of sampled-data systems with a linear periodic plant. II. H2-optimization of system St based on the Wiener–Hopf method. Autom. Remote Control 77(9), 1524–1543 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    V.G. Daniele, R. Zich, The Wiener–Hopf Method in Electromagnetics (SciTech Publishing, Chennai, 2014)CrossRefGoogle Scholar
  6. 6.
    C.H. Hsia, T.C. Wu, J.S. Chiang, A new method of moving object detection using adaptive filter. J. Real-Time Image Proc. 13(2), 311–325 (2017)CrossRefGoogle Scholar
  7. 7.
    R.W. Ibrahim, A. Ghani, Hybrid cloud entropy systems based on Wiener process. Kybernetes 45(7), 1072–1083 (2016)CrossRefGoogle Scholar
  8. 8.
    O.V. Lopushansky, A.V. Zagorodnyuk, Hilbert spaces of analytic functions of infinitely many variables. Ann. Polon. Math. 81(2), 111–122 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Lopushansky, Sectorial operators on Wiener algebras of analytic functions. Topology 48, 105–110 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S.S. Miller, P.T. Mocanu. Differential subordinations: theory and applications (CRC Press, Boca Raton, 2000)CrossRefGoogle Scholar
  11. 11.
    M. Rajni, S. Nagpal, V. Ravichandran. On a subclass of strongly starlike functions associated with exponential function. Bull. Malaysian Math. Sci. Soc. 38(1), 365–386 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Rabha W. Ibrahim
    • 1
  1. 1.Cloud Computing CenterUniversity of MalayaKuala LumpurMalaysia

Personalised recommendations