A unified approach to radius of convexity problems for certain classes of univalent analytic functions

We consider functions f analytic in the unit disc and assume the power series representation of the form f(z)=z+an+1zn+1+an+2zn+2+… where an+1 is fixed throughout. We provide a unified approach to radius convexity problems for different subclasses of univalent analytic functions. Numerous earlier estimates concerning the radius of convexity such as those involving fixed second coefficient, n initial gaps, n+1 symmetric gaps, etc. are discussed. It is shown that several known results, follow as special cases of those presented in this paper

Does practicing hatha yoga satisfy recommendations for intensity of physical activity which improves and maintains health and cardiovascular fitness?

Background: Little is known about the metabolic and heart rate responses to a typical hatha yoga session. The purposes of this study were 1) to determine whether a typical yoga practice using various postures meets the current recommendations for levels of physical activity required to improve and maintain health and cardiovascular fitness; 2) to determine the reliability of metabolic costs of yoga across sessions; 3) to compare the metabolic costs of yoga practice to those of treadmill walking. Methods: In this observational study, 20 intermediate-to-advanced level yoga practitioners, age 31.4 ± 8.3 years, performed an exercise routine inside a human respiratory chamber (indirect calorimeter) while wearing heart rate monitors. The exercise routine consisted of 30 minutes of sitting, 56 minutes of beginner-level hatha yoga administered by video, and 10 minutes of treadmill walking at 3.2 and 4.8 kph each. Measures were mean oxygen consumption (VO2), heart rate (HR), percentage predicted maximal heart rate (%MHR), metabolic equivalents (METs), and energy expenditure (kcal). Seven subjects repeated the protocol so that measurement reliability could be established. Results: Mean values across the entire yoga session for VO2, HR, %MHR, METs, and energy/min were 0.6 L/kg/min; 93.2 beats/min; 49.4%; 2.5; and 3.2 kcal/min; respectively. Results of the ICCs (2,1) for mean values across the entire yoga session for kcal, METs, and %MHR were 0.979 and 0.973, and 0.865, respectively. Conclusion: Metabolic costs of yoga averaged across the entire session represent low levels of physical activity, are similar to walking on a treadmill at 3.2 kph, and do not meet recommendations for levels of physical activity for improving or maintaining health or cardiovascular fitness. Yoga practice incorporating sun salutation postures exceeding the minimum bout of 10 minutes may contribute some portion of sufficiently intense physical activity to improve cardio-respiratory fitness in unfit or sedentary individuals. The measurement of energy expenditure across yoga sessions is highly reliable

Applications of Ruscheweyh derivatives and Hadamard product to analytic functions

For given analytic functions ϕ(z)=z+∑m=2∞ λm zm,ψ(z)=z+∑m=2∞ μm zm in U={z||z|<1} with λm≥0,μm≥0 and λm≥μm, let En(ϕ,ψ;A,B) be the class of analytic functions f(z)=z+∑m=2∞am zm in U such that (f*Ψ)(z)≠0 and Dn+1(f*ϕ)(z)Dn(f*Ψ)(z)≪1+Az1+Bz,      −1≤A<B≤1,  z∈U, where Dnh(z)=z(zn−1h(z))(n)/n!,   n∈N0={0,1,2,…} is the nth Ruscheweyh derivative; ≪ and * denote subordination and the Hadamard product, respectively. Let T be the class of analytic functions in U of the form f(z)=z−∑m=2∞am zm,  am≥0, and let En[ϕ,ψ;A,B]=En(ϕ,ψ;A,B)∩T. Coefficient estimates, extreme points, distortion theorems and radius of starlikeness and convexity are determined for functions in the class En[ϕ,ψ;A,B]. We also consider the quasi-Hadamard product of functions in En[z/(1−z),z/(1−z);A,B] and En[z/(1−z)2,z/(1−z)2;A,B]