On the minimum f-divergence for given total variation

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On Pinsker's and Vajda's Type Inequalities for Csiszár's $f$ -Divergences (2006, background): “Second, Vajda showed in [37] that the lower bound is convex, increasing and, for many ’s, satisfies that as (see also [38]).”
Information, Divergence and Risk for Binary Experiments (2009, result): “Gilardoni (2006b,c) presented a completely different approach which obtains many of the results of theorem 28.31 Gilardoni (2006c) improved Vajda’s bound slightly to KL(P,Q) ≥ ln 22−V − 2−V2 ln 2+V2 .”
Convex Functions: Constructions, Characterizations and Counterexamples (2010, background): “A remarkable application of the Lambert function to divergence estimates is given in [228].”
$f$ -Divergence Inequalities (2015, background): “The Rényi divergence of order α ≥ 0 from P to Q is given as follows: • If α ∈ (0, 1) ∪ (1,∞), then Dα(P‖Q) = 1 α− 1 log ( E [ exp ( α ıP‖Q(Y ) )]) (75) = 1 α− 1 log ( E [ exp ( (α− 1) ıP‖Q(X) )]) (76) with X ∼ P and Y ∼ Q. • If α = 0, then9 D0(P‖Q) = max F∈F : P (F)=1 log ( 1 Q(F) ) .”
On f-Divergences: Integral Representations, Local Behavior, and Inequalities (2018, unknown): “D], [60], [61], [62], [63]); 2) Inequalities which rely on the characterization of the exact locus of the joint range of f -divergences [34]; 3) f -divergence inequalities via functional domination ([58], [59, Section 3], [65], [66]); 4) Sharp f -divergence inequalities by using numerical tools for ”