We have extended the semianalytic technique of Iyer and Will for computing the complex quasinormal frequencies of black holes, ω, by constructing the Padé approximants of the (formal) series for ω 2 . It is shown that for the (so far best documented) quasinormal frequencies of the Schwarzschild and Reissner-Nordström black holes the Padé transforms P 6 6 and P 6 7 are, within the domain of applicability, always in excellent agreement with the numerical results. We argue that the method may serve as the black box with the "potential" Q(x)as an input and the accurate quasinormal modes as the output. The generalizations and modifications of the method are briefly discussed as well as the preliminary results for other classes of the black holes.

The first-order correction of the perturbative solution of the coupled equations of the quadratic gravity and nonlinear electrodynamics is constructed, with the zeroth-order solution coinciding with the ones given by Ay\'on-Beato and Garc{\'\i}a and by Bronnikov. It is shown that a simple generalization of the Bronnikov's electromagnetic Lagrangian leads to the solution expressible in terms of the polylogarithm functions. The solution is parametrized by two integration constants and depends on two free parameters. By the boundary conditions the integration constants are related to the charge and total mass of the system as seen by a distant observer, whereas the free parameters are adjusted to make the resultant line element regular at the center. It is argued that various curvature invariants are also regular there that strongly suggests the regularity of the spacetime. Despite the complexity of the problem the obtained solution can be studied analytically. The location of the event horizon of the black hole, its asymptotics and temperature are calculated. Special emphasis is put on the extremal configuration

The approximate renormalized stress-energy tensor of the quantized massive conformally coupled scalar field in the spacetime of electrically charged nonlinear black hole is constructed. It is achieved by functional differentiation of the lowest order of the DeWitt-Schwinger effective action involving coincidence limit of the Hadamard-Minakshisundaram-DeWitt-Seely coefficient a 3 . The result is compared with the analogous results derived for the ReissnerNordström black hole. It it shown that the most important differences occur in the vicinity of the event horizon of the black hole near the extremality limit. The structure of the nonlinear black hole is briefly studied by means of the Lambert functions.

The approximation of the renormalized stress-energy tensor of the quantized massive scalar, spinor, and vector field in Reissner-Nordström spacetime is constructed. It is achieved by functional differentiation of the lowest order of the Schwinger-DeWitt effective action involving the coincidence limit of the HadamardMinakshisundaram-DeWitt-Seely coefficient a 3 , and restricting the thus obtained general formulas to spacetimes with a vanishing curvature scalar. For the massive scalar field with an arbitrary curvature coupling, our results reproduce those obtained previously by Anderson, Hiscock, and Samuel by means of a sixth-order WKB approximation.

The near horizon limit of the extreme nonlinear black hole is investigated. It is shown that resulting geometry belongs to the AdS2 × S 2 class with different modules of curvatures of subspaces and could be described in terms of the Lambert functions. It is demonstrated that the considered class of Lagrangians does not admit solutions of the Bertotti-Robinson type.