Logarithmic Coefficient Bounds and Coefficient Conjectures for Classes Associated with Convex Functions

It is well-known that the logarithmic coefficients play an important role in the development of the theory of univalent functions. If S denotes the class of functions f z = z + ∑ n = 2 ∞ a n z n analytic and univalent in the open unit disk U , then the logarithmic coefficients γ n f of the function f ∈ S are defined by log f z / z = 2 ∑ n = 1 ∞ γ n f z n . In the current paper, the bounds for the logarithmic coefficients γ n for some well-known classes like C 1 + α z for α ∈ 0 , 1 and C V hpl 1 / 2 were estimated. Further, conjectures for the logarithmic coefficients γ n for functions f belonging to these classes are stated. For example, it is forecasted that if the function f ∈ C 1 + α z , then the logarithmic coefficients of f satisfy the inequalities γ n ≤ α / 2 n n + 1 , n ∈ ℕ . Equality is attained for the function L α , n , that is, log L α , n z / z = 2 ∑ n = 1 ∞ γ n L α , n z n = α / n n + 1 z n + ⋯ , z ∈ U .

On the third logarithmic coefficient in some subclasses of close-to-convex functions

For analytic functions f in the unit disk $${\mathbb {D}}$$D normalized by $$f(0)=0$$f(0)=0 and $$f'(0)=1$$f′(0)=1 satisfying in $${\mathbb {D}}$$D respectively the conditions $${{\,\mathrm{Re}\,}}\{ (1-z)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z+z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z)^2f'(z) \} > 0,$$Re{(1-z)f′(z)}>0,Re{(1-z2)f′(z)}>0,Re{(1-z+z2)f′(z)}>0,Re{(1-z)2f′(z)}>0, the sharp upper bound of the third logarithmic coefficient in case when $$f''(0)$$f′′(0) is real was computed.

ENTANGLEMENT ENTROPY FOR SPIN 2 FIELD ON A SHPERE

In this paper we compute the entanglement entropy for linearized gravitons (as helicity 2 particles) on a sphere defined over a Minkowski background. Previously, we analyse the cases of the solar and vector fields by decomposition in spherical harmonics. Then we generalise this method for the tensor field. We obtain the universal logarithmic coefficient and we analyse its relation to the conformal anomaly.

Myocardial protection by continuous, blood, antegrade-retrograde cardioplegia in rabbits

PURPOSE: To study the effectiveness of the continuous, blood, antegrade-retrograde cardioplegia in an experimental model of isolated heart, evaluating ventricular function. METHODS: Rabbits were divided into four groups: Control-C(n=10); ischemic crystalloid cardioplegia-IC(n=10); ischemic blood cardioplegia-IB(n=10); ischemic non cardioplegia-INC(n=10). After the ischemic protocol period the ventricular function was analyzed by the intra-ventricular balloon technique. RESULTS: the intra-ventricular developed pressure (IVDP) was: C(92.90±6.86mmHg); IC(77.78±6.15mmHg); IB(93.64±5.09mmHg); INC(39.46 ±8.91mmHg) p<0.005. The first derivative of intra-ventricular pressure in its positive deflection was: C(1137.50± 92.23mmHg/sec); IC(1130.62 ±43.78mmHg/sec); IB(1187.58± 88.38mmHg/sec); INC(620.02± 43.80mmHg/se) p<0.005. The first derivate pressure in its negative deflection was: C(770.00± 73.41mmHg/sec); IC(610.03 ±47.43mmg/sec); IB(762.53 ±46.02mmHg/sec); INC(412.35 ±84.36mmHg/sec) p<0,005. The stress-strain angular logarithmic coefficient was: C(0.108± 0.02); IC(0.159± 0.038); IB(0.114 ±0.016); INC(0.175± 0.038) p<0.05. CONCLUSION: The ischemic group protected by blood cardioplegia showed better ventricular function than ischemic group protected by crystalloid cardioplegia and the non protected group.

On the logarithmic coefficients of close-to-convex functions

Logarithmic coefficient bounds for some univalent functions are given in this paper.