Properties of certain complex integral operators

Abstract

Keywords: Alexander operator, α-Bloch space, β-Ces`aro operator, Bohr inequal ity, Boundary rotation, Bounded operator, Ces`aro operator, Close-to Convex functions, Compact operator, Convex functions, Essential norm, Hornich operations, Integral operator, Linear-invariant family, Locally univalent functions, Pre-Schwarzian norm, Radius of convexity, Separa ble space, Spectrum, Spirallike functions, Starlike and Univalent. The work of the whole thesis is based on the study of certain types of complex integral operators over analytic function spaces. These operators are obtained through the well known Hornich operations. These Hornich operations are frequently studied in Univalent function theory. In Chapter 1 we will have a descriptive note on these operations. This chapter also provides basic definitions of function spaces, properties, and some results which are useful in later chapters. One of the operators, namely β-Ces`aro operator, which we can obtain through the well-known Alexander operator and Hornich operations, which are studied over α-Bloch spaces in Chapter 2. In this chapter, we study the boundedness and compactness of β-Ces`aro operators. Moreover, with the help of the compactness property, we found the complete spectrum of these operators. We also have the Taylor series expansion of β-Ces`aro operators acting over bounded analytic functions. Therefore, we studied the Bohr phenomenon for the corresponding series representation of β-Ces`aro operators in Chapter 3 and similarly for other well-known integral operators. In Chapter 4 we have remarked on an open problem related to the univalency of the Hornich operations. Further, we establish the univalence properties of β-Ces`aro operators. Moreover, we calculated the Pre-Schwarzian norm of β-Ces`aro operators over the class of univalent functions. In this sequence, in Chapters 5 and 6 we study a more general operator which we obtain by the combination of Hornich operations and the Alexander operator. In Chapters 5 we find a subdisk of the unit disk such that the image of the subdisk under the integral operator is convex. In addition, we determine certain geometric properties such as convexity and close-to-convexity of the integral operators in Chapter 6