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<< Developer`
f[a_List, threshold_] := {First@a}~Join~
  PartitionMap[If[#[[2]] > #[[1]]*threshold, Mean@Drop[#, {2}], #[[2]]] &, a, 3, 1]
~Join~{Last@a}

f[a, 2]

{1, 1.2, 1.4, 1.1, 0.9, 0.9, 0.9, 1.1}

Employs the criterion $If[a_i>ba_{i-1},....]$, where threshold is the $b$.

f[a_List, threshold_] := {First@a}~Join~
  Developer`PartitionMap[
   If[#[[2]] > #[[1]]*threshold, Mean@Drop[#, {2}], #[[2]]] &, a, 3, 1]~Join~{Last@a}

f[a, 2]

{1, 1.2, 1.4, 1.1, 0.9, 0.9, 0.9, 1.1}

Employs the criterion $If[a_i>ba_{i-1},....]$, where threshold is the $b$.

<< Developer`
f[a_List, threshold_] := {First@a}~Join~
  PartitionMap[If[#[[2]] > threshold, Mean@Drop[#, {2}], #[[2]]] &, a, 3, 1]
~Join~{Last@a}

f[a, 2]

{1, 1.2, 1.4, 1.1, 0.9, 0.9, 0.9, 1.1}

One can of course make it as general as one wishes, including into the arguments of f the length of the partitions (3 here), offset (1 here), the conditions in If, what parts should be taken into account (the Drop part) etc. Above is an answer for this particular question to keep it simple.

<< Developer`
f[a_List, threshold_] := {First@a}~Join~
  PartitionMap[If[#[[2]] > #[[1]]*threshold, Mean@Drop[#, {2}], #[[2]]] &, a, 3, 1]
~Join~{Last@a}

f[a, 2]

{1, 1.2, 1.4, 1.1, 0.9, 0.9, 0.9, 1.1}

Employs the criterion $If[a_i>ba_{i-1},....]$, where threshold is the $b$.

<< Developer`
f[a_List, threshold_] := {First@a}~Join~
  PartitionMap[If[#[[2]] > threshold, Mean@Drop[#, {2}], #[[2]]] &, a, 3, 1]
~Join~{Last@a}

f[a, 2]

{1, 1.2, 1.4, 1.1, 0.9, 0.9, 0.9, 1.1}

<< Developer`
f[a_List, threshold_] := {First@a}~Join~
  PartitionMap[If[#[[2]] > threshold, Mean@Drop[#, {2}], #[[2]]] &, a, 3, 1]
~Join~{Last@a}

f[a, 2]

{1, 1.2, 1.4, 1.1, 0.9, 0.9, 0.9, 1.1}

One can of course make it as general as one wishes, including into the arguments of f the length of the partitions (3 here), offset (1 here), the conditions in If, what parts should be taken into account (the Drop part) etc. Above is an answer for this particular question to keep it simple.