Let f be analytic in D = {z : \z\ < 1} with f(O) = f'(O) - 1 = 0. For y > 0, the largest alpha(gamma) and beta(gamma) are found such that[GRAPHICS]The results solve the inclusion problem for convex and starlike functions defined in a sector.

Let K ( β ) K(\beta ) denote the class of normalized analytic strongly close-to-convex functions of order β ≥ 0 \beta \geq 0 , defined in the unit disc D D and let f ∈ K ( β ) f \in K(\beta ) , with f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ f(z) = z + {a_2}{z^2} + {a_3}{z^3} + \cdots , for z ∈ D z \in D . Sharp bounds are obtained for | a 3 − μ a 2 2 | |{a_3} - \mu a_2^2| when μ \mu is real.

Denote by A the class of functions f analytic in the unit disc D and normalised so that f(0) = f'(0)-1 = 0. For f is-an-element-of A and -1 < c less-than-or-equal-to 0, let F(c) be defined by F(c)(z) = (1 + c)/z(c) integral-0z t(c-1)f(t)dt for z is-an-element-of D. We find estimates on beta so that Re f'(z) > beta will ensure the starlikeness of F(c).

Forα>0, letB1(α)be the class of normalized analytic functions defined in the open unit discDsatisfyingRe(f(z)/z)α−1f′(z)>0forz∈D. The sharp lower bound forRe(f(z)/z)αis obtained and the result is generalized to some iterated integral operators.

The UK approach to pollution control in tidal waters places considerable emphasis on balancing the contrasting needs of utilization of the waters for the assimilation of effluents with the protection of all identified uses, i.e. the Environmental Quality Objective/Environmental Quality Standard (EQO/EQS) approach. In order to protect uses of the waters for bathing and general amenity, this approach in the design of improvement schemes necessary for marine sewage discharges is examined. However, recent guidelines for consent applications dealing with schemes designed to protect bathing use suggest a departure from the EQO/EQS approach towards an emission standard when dealing with the solid phase of the sewage effluent. The implications of these approaches are discussed from the design (scientific and engineering) viewpoints. A summary of headworks processes and the current practice for preliminary treatment at marine outfall sites in the UK is reviewed, together with comments on their effectiveness. The terms of reference and work of a joint working group to examine the efficiency of screening equipment at marine outfall headworks is described, together with a technical summary of results collected at a range of sites and with various types of equipment. Recommendations for future research are given, and a strategy for screening is developed.

For α > 0 \alpha {\text { > }}0 , let B 1 ( α ) {B_1}(\alpha ) be the class of normalised analytic functions f f defined in the open unit disc D D such that Re( f ( z ) / z ) α − 1 f ′ ( z ) > 0 {\text {Re(}}f(z)/z{{\text {)}}^{\alpha - 1}}f’(z){\text { > }}0 for z ∈ D z \in D . Sharp upper and lower bounds are obtained for | z f ′ ( z ) / f ( z ) | \left | {zf’(z)/f(z)} \right | when f ∈ B 1 ( α ) f \in {B_1}(\alpha ) .

For α>0, let B(α) be the class of regular normalized Bazilevič functions defined in the unit disc. Choosing the associated starlike function g(z)≡z gives a proper subclass B1(α) of B(α). For B(α), correct growth estimates in terms of the area function are unknown. Several results in this direction are given for B1(12).

Let B(α) be the class of normalised Bazilevič functions of type α>0 with respect to the starlike function g. Let B1(α) be the subclass of B(α) when g(z)≡z. Distortion theorems and coefficient estimates are obtained for functions belonging to B1(α).

Do Sedge Warblers view 'habitat' in terms of plant species, or of other aspects of vegetation structure such as its density?

In this paper, a new class of normalized univalent functions is introduced. The properties of this class and its relationship with some other subclasses of univalent functions are studied. The functions in this class are close-to-convex.

Autumn migrant Garden Warblers in southern Portugal were found feeding extensively on figs, and such birds were heavier than others taking alternative foods. Hence figs were considered important for the rapid deposition of migratory fat, required for the Saharan crossing.

The paper is concerned with the integral \[ H = ∫ 0 2 π | f | σ | F | τ ( Re ⁡ F ) κ d θ H = \int _0^{2\pi }|f{|^\sigma }|F{|^\tau }{(\operatorname {Re} F)^\kappa }\;d\theta \] in which f is a function regular and starlike in the unit disc, F = z f ′ / f F = zf’/f , and the parameters σ , τ , κ \sigma ,\tau ,\kappa are real. A study of H is of interest since various well-known integrals in the theory, such as the length of f ( | z | = r ) f(|z| = r) , the area of f ( | z | ⩽ r ) f(|z| \leqslant r) , and the integral means of f, are essentially obtained from it by suitably choosing the parameters. An asymptotic formula, valid as r → 1 r \to 1 , is obtained for H when f is a starlike function of positive order α \alpha , and the parameters satisfy α σ + τ + κ > 1 , τ + κ ⩾ 0 , κ ⩾ 0 , σ > 0 \alpha \sigma + \tau + \kappa > 1,\tau + \kappa \geqslant 0,\kappa \geqslant 0,\sigma > 0 . Several easy applications of this result are made; some to obtaining old results, two others in proving conjectures of Holland and Thomas.

Journal of the London Mathematical SocietyVolume s1-42, Issue 1 p. 427-435 Notes and papers On Starlike and Close-to-Convex Univalent Functions D. K. Thomas, D. K. Thomas Department of Mathematics, Imperial College LondonSearch for more papers by this author D. K. Thomas, D. K. Thomas Department of Mathematics, Imperial College LondonSearch for more papers by this author First published: 1967 https://doi.org/10.1112/jlms/s1-42.1.427Citations: 17AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes1-42, Issue11967Pages 427-435 RelatedInformation

AbstractA number of phenols, simple and N‐substituted aromatic amines, thioamides, and thioanilides which are known to inhibit free radical reactions have been introduced into dicumyl peroxide–natural rubber cures, and their influence upon the degree of crosslinking has been determined. A simple kinetic scheme allows a calculation of the relative reactivities of these additives and the values obtained are compared with those of other workers for the same additives but in different systems. It is concluded that the additives interfere with the crosslinking reaction by exchanging a hydrogen atom with the free radicals involved, and the sequence of reactivities is readily explained in terms of current theories for the effect of substituents upon the reactivity of organic compounds. Types of materials likely to be useful as antioxidants in dicumyl cures are discussed.

The exposure of polymethylmethacrylate ( PMM ) and many other macromolecules to high-energy y- or electron radiation produces a degradation of the molecular chains. This may result from either ionization or excitation. Ultra-violet light results in excitation only. This paper compares the effects of these forms of radiation. The degradation of PMM has been studied in thick or thin films, and in solutions in benzene. It is found that at room temperature degradation (random main chain fracture) occurs with ultra-violet radiation, whereas at higher temperatures the reaction is one of chain depolymerization. The energy absorbed per main chain fracture is about 550 eV with ultra ­ violet radiation, so that less than 1 % of the quanta absorbed are effective; this compares with about 65 eV of γ-energy absorbed per fracture. Apart from this difference in efficiency, the two reactions are very similar. In both cases the number of fractures is proportional to dose and the radicals formed (as seen by e.s.r.) are identical in character. Somewhat similar changes also appear in the optical spectrum, although these may be modified by surface oxygen. These results show no significant contribution from any ionization produced by high-energy radiation.

D. K. Thomas, Trans. Faraday Soc., 1961, 57, 511 DOI: 10.1039/TF9615700511

AbstractThe intrinsic viscosities of a series of polyethylene glycols have been measured in water and toluene, and a relationship proposed between molecular weight and intrinsic viscosity. Values obtained for the Huggins constant below a molecular weight of 4000 are much larger than would be anticipated for linear high polymers. Some information about the variation of configuration with molecular weight has been obtained by comparing the specific viscosities with those expected from the relationships of Einstein and Guth and Gold for spherical particles, and those of Kuhn and Simha for rodlike particles.

Measurements of the equivalent conductance of sulphuric, nitric, and hydrofluoric acids were made in water – hydrogen peroxide mixtures by means of a direct-current method. It was found that in solvents containing a high percentage of hydrogen peroxide the conductivities of these acids were extremely low, and it is proposed that this results from a marked decrease in their degrees of dissociation. This proposal is supported by similar measurements made on the system water – strong acid – weak acid.