The convolution of convex harmonic univalent functions in the unit disk, unlike analytic functions, may not be or even univalent. main purpose this work is to develop previous in...

‎In this paper,  the sufficient conditions for the linear combinations of slanted half-plane harmonic mappings to be univalent and convex in the direction of $(-gamma) $ are studied. Our result improves some recent works. Furthermore, a illustrative example and imagine domains of the linear combinations satisfying the desired conditions are enumerated.

In this paper, we mainly consider the convolutions of slanted half-plane mappings and strip unit disk D. If f1 is a mapping f2 or mapping, then prove that * convex in some direction if locally univalent We also obtain two sufficient conditions for to be Our results extend many recent direction. Moreover, class harmonic including mappings, as consequence, any combination such sense-preserving co...

If a pattern of concentric circles, interrupted so as to produce the perception of a subjective bar extending from the centre to the periphery of the pattern, was slowly rotated in a plane perpendicular to the line of sight, observers reported seeing the bar slanted in depth and moving over complete and stationary concentric circles. When the interrupted concentric circles were completed by red...

We consider the class of univalent log-harmonic mappings on unit disk. Firstly, we present general idea constructing Koebe mappings, right half-plane and two-slits then show precise ranges these mappings. Moreover, coefficient estimates for starlike are obtained. Growth distortion theorems certain special subclasses studied. Finally, propose two conjectures, namely, covering conjectures.

The perceived depth of features is known to be affected by the presence of a slanted reference plane. Mitchison and Westheimer ((1984). Vision Research, 24, 1063-1070) reported that two lines appear to be at the same depth when they lie in a plane approximately parallel with the reference plane. We measured the perceived depth of two lines presented in front of a regular grid of dots that was e...