The main aim of this paper is to use the method of differential subordination to obtain a number of sufficient conditions for a normalized analytic function to be univalent or starlike in the unit disc. In particular, we find a condition on β so that each normalized analytic function f satisfying the condition ∣∣∣1 + zf ′′(z) 2f ′(z) − zf ′(z) f(z) ∣∣∣ < β, z ∈ Δ implies that f is univalent or ...

This note is a sequel to the paper On rings on rings by Anatole Beck [ l ] . The problem we consider (originally proposed by Paul Rosenbloom) is that of characterizing the parameter p for the annulus Q = {1< | s| <p} in terms of the ring R of bounded analytic functions nn this annulus. Beck's solution involves properties of univalent functions, and although the subset of univalent functions in ...

The univalent hapten, nonadeca lysyl epsilon-Dnp-lysine, binds tightly to rabbit anti-2,4-dinitrophenyl antibody, and the complex has a sedimentation coefficient of 6.7, characteristic of a single antibody molecule. In this communication, we show that this complex is a good activator of the serum complement system. For activation to occur, the univalent hapten must contain the specific group wh...

Spinach chloroplasts induced the photoreduction of cytochrome c and photooxidation of epinephrine, both of which depended on oxygen and were inhibited by superoxide dismutase. This confirmed the univalent reduction of molecular oxygen in illuminated chloroplasts. In the presence of cytochrome c, hydrogen peroxide formation by chloroplasts was inhibited completely. Rates for the photoreduction o...

Let A denote the family of all analytic functions f in the unit disk D with the normalization f (0) = 0 = f ′(0)− 1. Define S = { f ∈ A : f is univalent in D}, U = { f ∈A : ∣∣ f ′(z)(z/ f (z))2−1∣∣< 1 for z ∈ D}, and P(1/2) = { f ∈A : Re( f (z)/z) > 1/2 for z ∈ D}. In this paper, we determine the radius of univalency of F(z) = z f (z)/g(z) whenever f ∈S or U , and g ∈S or P(1/2). Based on our i...

In [6] it was shown that if f ∈ is starlike of order α, α= 0.294, . . . , so is the Libera integral operator F . We also know that (see, e.g., [1]) there are functions which are univalent or spiral-like in so that their Libera integral operators are not univalent or spiral-like in . Li and Owa [5] proved that if f ∈ is univalent in , then Fn(z) is starlike in |z| < 3/8. The number 3/8 is sharp....

Abstract. In this paper we define and verify a subclass of harmonic univalent functions involving the argument of complex-value functions of the form f = h + ¯g and investigate some properties of this subclass e.g. necessary and sufficient coefficient bounds, extreme points, distortion bounds and Hadamard product.Abstract. In this paper we define and verify a subclass of harmonic univalent func...

Treatment of 6.5-S rabbit antibody with pepsin removes a large inactive fragment. The residual portion of the molecule, which is still bivalent, has a molecular weight of 106,000 and a sedimentation coefficient of 4.6 to 5.0 S (1, 2). Reduction of a single, highly labile disulfide bond cleaves this fragment approximately in half (2, 3). The resulting 3.5-S fragments are univalent and inhibit sp...

While working on the completion of the proof of the Bloch-Kato conjecture I have thought a lot about what to do next. Eventually I became convinced that the most interesting and important directions in current mathematics are the ones related to the transition into a new era which will be characterized by the widespread use of automated tools for proof construction and verification. I have star...