Sharp Coefficient Research Articles

Coefficients and Integral Means of Some Classes of Analytic Functions

The sharp coefficient bounds for the classes ${V_k}$ of functions of bounded boundary rotation are obtained by a short and elementary argument. Elementary methods are also applied for the coefficients of related classes characterised by a generalised Kaplan condition. The result ${(1 + xz)^\alpha }{(1 - z)^{ - \beta }} \ll {(1 + z)^\alpha }{(1 - z)^{ - \beta }}$ $(\left | x \right | = 1,\alpha \geqslant 1,\beta \geqslant 1)$ is proved simply. It is further shown that the functions ${(1 + z)^\alpha }{(1 - z)^{ - \beta }}$ are extremal for the $p$th means ($p$ an arbitrary real) of all Kaplan classes $K(\alpha ,\beta )$.

Coefficients and integral means of some classes of analytic functions

The sharp coefficient bounds for the classes V k {V_k} of functions of bounded boundary rotation are obtained by a short and elementary argument. Elementary methods are also applied for the coefficients of related classes characterised by a generalised Kaplan condition. The result ( 1 + x z ) α ( 1 − z ) − β ≪ ( 1 + z ) α ( 1 − z ) − β {(1 + xz)^\alpha }{(1 - z)^{ - \beta }} \ll {(1 + z)^\alpha }{(1 - z)^{ - \beta }} ( | x | = 1 , α ⩾ 1 , β ⩾ 1 ) (\left | x \right | = 1,\alpha \geqslant 1,\beta \geqslant 1) is proved simply. It is further shown that the functions ( 1 + z ) α ( 1 − z ) − β {(1 + z)^\alpha }{(1 - z)^{ - \beta }} are extremal for the p p th means ( p p an arbitrary real) of all Kaplan classes K ( α , β ) K(\alpha ,\beta ) .

研削砥石のドレッシングに関する研究 (第1報)

A new dressing method is introduced, in which. a single point diamond dresser is mounted on eccentrical rotating spindle and fed parallel to the wheel surface, for higher performance of grinding wheel. It is expected in this method that cutting edges on the wheel surface are sharpened by more chance of cleavage fracture, because grains are forced in various directions. Sharpness coefficient of cutting edge, estimated initial wear area and mean number of cutting edges are defined ascharacteristic values to quantify the condition of cutting edges on the wheel surface. The dressing process is significantly affected by the revolution ratio of dresser to wheel and the decimal place of the ratio should be selected except 0 and 0.5. Cutting edges generated in the new method are sharper, while the number of cutting edges is slightly fewer than in conventional one. Grinding force and residual stock are smaller while surface roughness and wheel wear are larger in this method and grinding ability is maintained longer as compared with conventional one.

Estimates for inverse coefficients of univalent functions from integral means

The purpose of this note is to point out that sharp coefficient bounds for the inverses of univalent functions from certain families are fairly direct corollaries of results on integral means. As an example, in §1 the method is applied to the familiar schlicht classS. The resulting coefficient estimates for the inverses of functions inS were first obtained by K. Lowner. Following this prototype, in §2 we obtain corresponding results, which are new, for a classS(p) of meromorphic schlicht functions in |z|<1.

Distortion Properties of Holomorphic Functions of Several Complex Variables

Let Q(D) be a class of functions q, q(0) = 0, |q(z)| < 1 holomorphic in the Reinhardt domain D ⊂ Cn, a and b — arbitrary fixed numbers satisfying the condition — 1 ≤ b < a ≤ 1. 𝔭(a, b; D) — the class of functions p such that p ϵ 𝔭(a, b; D) iff for some q ϵ Q(D) and every z ϵ D. S*(a, b; D) — the class of functions f such that f ϵ S*(a, g; D) iff Sc(a, b; D) — the class of functions q such that q ϵ Sc(a, b; D) iff , where p e 𝔭(a, b; D) and K is an operator of the form for z=z1,z2,…zn. The author obtains sharp bounds on |p(z)|, f(z)| g(z)| as well as sharp coefficient inequalities for functions in 𝔭(a, b; D), S*(a, b; D) and Sc(a, b; D).

Coefficient estimates for starlike functions

Let (α β) denote the class of functionsanalytic in the unit disc Δ ≡{z: |z| &lt; 1} and satisfyingfor some α, β (0 ≤ α &lt; 1, 0 &lt; β ≤ 1) and for all z ∈ Δ. In the present paper, sharp coefficient estimates for functions in (α, β) have been obtained. The results thus obtained not only generalize the corresponding results of Thomas H. MacGregor (Michigan Math. J. 10 (1963), 277–281), A.V. Boyd (Proc. Amer. Math. Soc. 17 (1966), 1016–1018) and others, but also give rise to analogous results for various other subclasses of starlike functions.

A Class of Univalent Functions

A sharp coefficient estimate is obtained for a class $D(\alpha )$ of functions univalent in the open unit disc. The radius of convexity and an arclength result are also determined for the class.

A class of univalent functions

A sharp coefficient estimate is obtained for a class D ( α ) D(\alpha ) of functions univalent in the open unit disc. The radius of convexity and an arclength result are also determined for the class.

On the coefficients of Bazilevič functions

Let B ( α ) B(\alpha ) denote the class of normalized ( f ( 0 ) = 0 , f ′ ( 0 ) = 1 ) (f(0) = 0,f’(0) = 1) , Bazilevič functions of type α \alpha defined in Δ : | z | &gt; 1 , i.e. f ( z ) = [ α ∫ 0 z P ( ζ ) g ( ζ ) α ζ − 1 d ζ ] 1 / α \Delta :|z| &gt; 1,{\text {i.e.}}f(z) = {[\alpha \smallint _0^zP(\zeta )g{(\zeta )^\alpha }{\zeta ^{ - 1}}d\zeta ]^{1/\alpha }} where g ( ζ ) g(\zeta ) is starlike in Δ , P ( ζ ) \Delta ,P(\zeta ) is regular with ℜ P ( ζ ) &gt; 0 \Re \,P(\zeta ) &gt; 0 in Δ \Delta and α &gt; 0 \alpha &gt; 0 . Let B m ( α ) {B_m}(\alpha ) denote the subclass of B ( α ) B(\alpha ) which is m-fold symmetric ( f ( e 2 π i / m z ) = e 2 π i / m f ( z ) , m = 1 , 2 , ⋯ ) (f({e^{2\pi i/m}}z) = {e^{2\pi i/m}}f(z),m = 1,2, \cdots ) . Functions in B ( α ) B(\alpha ) have been shown to be univalent. The authors obtain sharp coefficient inequalities for functions in B m ( 1 / N ) {B_m}(1/N) where N is a positive integer. In addition an example of a Bazilevič function which is not close-to-convex is given.