The most general formulation of Penrose's inequality yields a lower bound for Arnowitt-Deser-Misner mass in terms of the area, charge, and angular momentum of black holes. This inequality is in turn equivalent to an upper and lower bound for the area in terms of the remaining quantities. In this paper, we establish the lower bound for a single black hole in the setting of axisymmetric maximal initial data sets for the Einstein-Maxwell equations, when the non-electromagnetic matter fields are not charged and satisfy the dominant energy condition. It is shown that the inequality is saturated if and only if the initial data arise from the extreme Kerr-Newman spacetime. Further refinements are given when either charge or angular momentum vanish. Last, we discuss the validity of the lower bound in the presence of multiple black holes.

Aims: To examine the availability of frailty concept with objective criteria for risk stratification in patients with congestive heart failure (CHF). Methods and results: Study design was secondary analysis of our CHF cohort. We selected 181 patients who completed clinical assessments and were successfully followed 2-year post discharge. To set frailty criteria, grip strength <26 kg in men and <17 kg in women (weakness) and performance measure for activities of daily living-8 ≧21 points (exhaustion) were defined for predicting 6 min walking distance <300 m (slowness) by the receiver-operating characteristics. During 2 years of follow up, subjects who met all the criteria had a 4 times greater risk of cardiac event compared with those with no frailty criteria. Conclusion: The findings of present study suggest that frailty criteria may serve as a new clinical marker for management of patients with CHF.

Background: In patients with heart failure (HF), physical frailty should be assessed to enable risk stratification. No conventional frailty criteria have so far been developed considering HF‐specific outcomes. This study aimed to propose a frailty‐based prognostic score using a nationwide cohort study of Japanese patients with HF.Methods: We analysed 2721 patients hospitalized for HF and capable of walking at discharge (median age: 76 years [interquartile range 67–83], men: 60.5%). Physical frailty was evaluated at discharge using four quantitative measures: usual walking speed, grip strength, Performance Measure for Activities of Daily Living‐8 (PMADL‐8), and Self‐Efficacy for Walking‐7 (SEW‐7). The primary outcome was a composite of HF rehospitalization and all‐cause mortality within 2 years. A cut‐off point was identified for each measure using receiver operating characteristic analysis in a derivation cohort (n = 1778). Cox proportional hazards model was used to assign a score to each frailty domain according to the correlation with the endpoint. Patients were divided into four categories according to the sum score, and survival was compared by analysing the Kaplan–Meier curve and Cox proportional hazards model. Cumulative incidences of the events according to frailty categories were compared between the derivation cohort and a validation cohort (n = 943).Results: The cut‐off value and assigned score of each indicator was determined as follows: usual walking speed < 0.98 m/s = 4 points; grip strength < 30.0 kg (men) or 17.5 kg (women) = 5 points; PMADL‐8 ≥ 21 points = 2 points; SEW‐7 ≤ 20 points = 3 points. We stratified patients into four categories according to the sum score: Category I, ≤3 points; Categories II, 4–8 points; Category III, 9–13 points; and Category IV, 14 points. The prevalence and cumulative incidence of the composite outcome for Categories I to IV in the derivation cohort were 27.4%, 25.2%, 26.4%, and 21.0%, and 9.5, 16.3, 26.3, and 36.8/100 person‐years, respectively. Similar results were confirmed in the validation cohort. In Cox proportional hazards model, frailty categories were associated with the composite outcome independent of potential confounders (hazard ratio [95% confidence interval] in reference to Category I: Categories II, 1.51 [0.84–2.72], P = 0.169; Category III, 2.37 [1.32–4.23], P = 0.004; Category IV, 2.66 [1.45–4.89], P = 0.002).Conclusions: The frailty‐based prognostic score proposed in this study was well associated with prognosis and will serve for risk stratification in patients with HF.